Quantitative estimates for the flux of TASEP with dilute site disorder
C. Bahadoran$^a$
T. Bodineau$^b$
C. Bahadoran$^1$
T. Bodineau$^2$
We prove that the flux function of the totally asymmetric simple exclusion process (TASEP) with site disorder exhibits a flat segment for sufficiently dilute disorder. For high dilution, we obtain an accurate description of the flux. The result is established under a decay assumption of the maximum current in finite boxes, which is implied in particular by a sufficiently slow power tail assumption on the disorder distribution near its minimum. To circumvent the absence of explicit invariant measures, we use an original renormalization procedure and some ideas inspired by homogenization.
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%\title[Flux of TASEP with site disorder]{Quantitative estimates for the flux of TASEP with dilute site disorder}
%\author{C. Bahadoran$^a$, T. Bodineau$^b$}
%$$ \ba{l}
%^a\,\mbox{\small Laboratoire de Math\'ematiques Blaise Pascal, Universit\'e Clermont Auvergne, 63177 Aubi\`ere, France} \\
%\quad \mbox{\small e-mail:
%bahadora@math.univ-bpclermont.fr}\\
%%
%^b\,\mbox{\small Laboratoire de Math\'ematiques Blaise Pascal, Universit\'e Clermont Auvergne, 63177 Aubi\`ere, France} \\
%\quad \mbox{\small e-mail:
%bahadora@math.univ-bpclermont.fr}
%%
%\ea
%$$
%\thanks{We would like to thank A. Sznitman for very useful discussions. We acknowledge support of grant ANR-15-CE40-0020-02}
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\begin{document}
%
\title[Flux of TASEP with site disorder]{Quantitative estimates for the flux of TASEP with dilute site disorder}
\author{C. Bahadoran$^1$, T. Bodineau$^2$}
%\date{\today}
\thanks{We would like to thank A. Sznitman for very useful discussions. We acknowledge support of grant ANR-15-CE40-0020-02}
\maketitle
$$ \ba{l}
^1\,\mbox{\small Laboratoire de Math\'ematiques Blaise Pascal, Universit\'e Clermont Auvergne, 63178 Aubi\`ere, France} \\
\quad \mbox{\small e-mail:
bahadora@math.univ-bpclermont.fr}\\
%
^2\,\mbox{\small CMAP, Ecole Polytechnique, CNRS, Universit\'e Paris-Saclay, 91128 Palaiseau, France} \\
\quad \mbox{\small e-mail:
thierry.bodineau@polytechnique.edu}
%
\ea
$$
%
%
\begin{abstract}
We prove that the flux function of the totally asymmetric simple exclusion process (TASEP) with site disorder exhibits a flat segment for sufficiently dilute disorder. 
For high dilution, we obtain an accurate description of the flux.
The result is established under
a decay assumption of the maximum current in finite boxes, which is implied in particular by a sufficiently slow power tail assumption on the disorder distribution near its minimum.
To circumvent the absence of explicit invariant measures, we use an original renormalization procedure and some ideas inspired by homogenization.
\end{abstract}
%
\noindent\textbf{MSC:} 60K35, 60K37, 82C22\\ \\
%
\textbf{Keywords:} Totally asymmetric exclusion process; site disorder; flux function; plateau; phase transition; renormalization; homogenization





%\tableofcontents




\section{Introduction}
%
%{\color{black} il faudrait citer \cite{GKS}}
The flux function, also called current-density relation in traffic-flow physics \cite{css}, is the most fundamental object to describe the macroscopic behavior of driven lattice gases.
%, as well as, to some extent, their fluctuations around this mean behavior. 
The paradigmatic model in this class is the totally asymmetric simple exclusion process (TASEP), where particles on the one-dimensional integer lattice hop to the right at unit rate and obey an exclusion rule.
%independently except that jumps are constrained to respect the exclusion rule. 
Density $\rho\in(0,1]$ is the only conserved quantity and is associated locally to a 
%its local density $\rho\in(0,1]$  has an associated 
flux (or current) that is defined as the amount of particles crossing a given site per unit time in a system with homogeneous density $\rho$. For TASEP, the flux function is explicitly given by
%
\be\label{flux_intro}
 f(\rho)=\rho(1-\rho).
\ee
%
In the hyperbolic scaling limit \cite{rez}, the empirical particle density field is governed by entropy solutions of the scalar conservation law
%
\be
\label{burgers_intro}
\partial_t\rho(t,x)+\partial_x  \, f \big( \rho(t,x) \big)=0,
\ee
%
with $f$ given by \eqref{flux_intro}. 
%
%This flux function is obtained as a steady state expectation of the microscopic flux:
%%
%\be\label{def_flux_intro}
%f(\rho):=\int j_x(\eta)d\nu_\rho(\eta),
%\ee
%%
%where
%%
%\be
%\label{flux_micro_intro}
%j_x(\eta):=\eta(x)[1-\eta(x+1)].
%\ee
%%
%In \eqref{flux_micro_intro}, $\eta$ denotes a particle configuration and $\eta(x)\in\{0,1\}$ the number of particles at site $x$. The microscopic flux \eqref{flux_micro_intro}  of TASEP is simply
%the rate at which a particle hops to the right, and $\nu_\rho$ denotes the product Bernoulli
%measure $\nu_\rho$ with parameter $\rho\in[0,1]$, that is extremal invariant for TASEP.
This kind of result can be extended to a variety of asymmetric models \cite{sep, rez2,bgrs4}, 
but  when the invariant measures are not explicit, little can be said about the flux function. 
Nevertheless, convexity or concavity can be obtained as a byproduct of the variational approach set up in \cite{sep0}, which applies to  totally asymmetric models with state-independent jump rates, like TASEP. However strict convexity or strict concavity, which are related to the absence of a phase transition, require new mathematical ideas to be derived for general driven dynamics.


%In fact, one does not even know whether an extremal measure exists for every possible density value. Nevertheless, 
%%the approaches of \cite{rez2} and \cite{bgrs2} show that 
%the flux function can still be expressed by \eqref{def_flux_intro}--\eqref{flux_micro_intro}
%for {\color{black} some} densities, and the existence of a gap in  {\color{black} a density} interval implies that the flux in this interval is necessarily defined by linear interpolation \cite{rez2, bgrs4}.
%%For existing density values, the invariant measure is known to be spatiallt ergodic (\cite{sepbook}).
%%
%The latter case corresponds to a phenomenon of phase segregation at mesoscopic scale: if the local hydrodynamic density lies in a density gap, the system cannot reach a local equilibrium 
%state at the corresponding density. Instead, the system reorganizes itself into a succession of mesoscopic stretches of lower and higher densities {\color{black} associated} to the closest steady states. 
%{\color{black} These mesoscopic density variations} 
%average out in a small
%macroscopic box to yield the hydrodynamic density. Conversely, the presence of a partial linear portion on the flux function suggests a phase transition. Although such a phenomenon is not believed to happen for homogenous systems,
%one has neither a proof nor a counterexample.\\ 


\medskip

In disordered systems, a phase transition has been first proved  for nearest-neighbor asymmetric site-disordered zero-range processes (ZRP) and its signature is a constant flux on a density interval $[\rho_c,+\infty)$, where $\rho_c$ is the density of the maximal invariant measure. 
The invariant measures  of  the disordered ZRP are explicit so that the flux can be exactly computed and the phase transition precisely located.
The necessary and sufficient condition for the occurrence of a phase transition is a slow enough tail of  the jump rate distribution near its minimum value $r$. Microscopically,  phase transition takes the form of Bose-Einstein condensation \cite{ev}. 
%In a finite system of size $L$ with periodic boundary conditions,
%a stationary state with mean density $\rho>\rho_c$ is achieved by completing the background state of density $\rho_c$ with a macroscopic condensate of mass $L(\rho-\rho_c)$ at the slowest site. 
In an infinite system with mean drift to the right, the excess mass is captured by the asymptotically slowest sites at $-\infty$. This was 
proven rigorously in \cite{afgl} for the totally asymmetric ZRP with constant jump rate with respect to the number of particles which is equivalent to TASEP with {\it particlewise} disorder \cite{ks}. This holds also  for nearest-neighbor ZRP with more general  jump rates \cite{bmrs} (see also \cite{fs} for partial results in higher dimension).
% and shown to fail for non-nearest neighbor zero-range processes. 
The TASEP picture can be interpreted as a traffic-flow model with slow and fast vehicles. The phase transition then occurs on a density interval $[0,\rho_c]$, where the flux is linear with a slope equal to the constant mean velocity of the system. This velocity is imposed by the slowest vehicles at $+\infty$. As one moves ahead, slower an slower vehicles are encountered, followed by a platoon of faster vehicles, and preceded by a gap before the next platoon \cite{ks}. 
%The platoon is at equilibrium with
%critical density $\rho_c$, while the gap is the TASEP version of the condensate in the ZRP picture. 
%%At high densities $\rho\geq\rho_c$, the exclusion rules becomes dominant with respect to disorder, and the flux has  nonlinear concave shape. 
%The hydrodynamic limit for disordered asymmetric ZRP was first established in  \cite{bfl}
%in any space dimension but only in the subcritical density interval. The full hydrodynamic limit (including the plateau) was proved in \cite{ks}
%for the TAZRP with constant rate, but remains open for more general ZRP. \\ 


\medskip


In this paper, we consider TASEP with i.i.d. site disorder such that the jump rate at each site has a random value
whose distribution is supported in an interval $[r,1]$, with $r\in(0,1)$. 
A flat piece in the flux was observed numerically by physicists  \cite{kru, tb, HS} and interpreted as
the occurence of a phase transition by several heuristic arguments.
Contrary to the disordered ZRP, the invariant measures are no longer explicit in the site-disordered TASEP, which makes the analysis of the flux more challenging.
Before commenting on the  flat segment in the flux, let us mention that 
 the existence of a hydrodynamic limit of the form \eqref{burgers_intro} for TASEP with i.i.d. site disorder was established in \cite{sep}, using last passage percolation (LPP) and variational coupling.
Consequently, the flux function was shown to be concave. More generally, the existence of a limit of the type \eqref{burgers_intro} was obtained in \cite{bgrs4}
for asymmetric attractive systems in ergodic environment, based on the study of invariant measures.
We refer also to \cite{CL, sch1, sch2, sz} for further rigorous results in a different class of disordered SEP.





 \medskip
 

Recently, Sly gave in \cite{sly}  a short and very elegant proof of the existence of a flat segment 
in the flux for TASEP with general rate distribution. 
The proof in \cite{sly} relies on a clever coupling implemented in the LPP formulation of the TASEP.
In this paper, we develop a different approach, announced in \cite{bb}, based on a renormalization 
method to obtain a precise information on the flux function and on the flat segment  at the price  of 
additional assumptions on the disorder distribution.
To our knowledge, this is the first time a renormalization procedure is applied in the context of TASEP.
We focus on the case of dilute disorder which plays a key role in the physical literature  \cite{kru}.
The jump rate at each site is chosen randomly, according to some dilution parameter $\varepsilon\in[0,1]$, 
so that a site is ``fast'' with probability $1-\varepsilon$,
in which case it has rate $1$, or ``slow'' with probability $\varepsilon$, in which case its rate has some distribution $Q$ with support $(r,1]$ for some $r\in(0,1)$. Under some assumption on the distribution $Q$, and 
for sufficiently diluted disorder, i.e. $\varepsilon$ small enough, we prove the existence of a flat segment 
and determine the limiting size of this segment when 
$\varepsilon$ vanishes. Moreover, we prove the convergence of the whole flux function to  an explicit function.
%{\color{black} In particular, we get a precise control of the size of the flat segment.}
We stress the fact that  Sly's argument \cite{sly} does not require any assumption on $Q$ nor the dilution of the disorder, however the control on the flux in \cite{sly} is less precise.





\medskip


The physical interpretation of the flat segment in the flux \cite{kru} is 
the emergence at different scales of atypical disorder slowing down the particles and leading to traffic jams.
As one moves ahead along the disorder, slowest and slowest regions are encountered, with larger and larger stretches of sites with the minimal rate $r$ (or near this minimal rate). Locally a slow stretch of environment inside a typical region is expected to create a
picture similar to the {\it slow bond} TASEP introduced in \cite{jl}. It is known that a slow bond with an arbitrarily small blockage \cite{bss} restricts the local current. On the hydrodynamic scale \cite{sepslow}, this creates a traffic jam
with a high density of queuing vehicles to the left and a low density to the right, that is an antishock for Burgers' equation.
%
Renormalization turns the problem into a hierarchy of slow-bond like pictures, where at each scale, the difference between 
the ``typically fast'' and ``atypically slow'' region becomes smaller and smaller.
Slower jams will gradually absorb faster ones so that one expects to see a succession of mesoscopically growing shocks and antishocks.
Some results in this direction were obtained in \cite{GKS} in the case of particle disorder.
%
%
%{\color{black}
Even though, a single slow bond induces a phase transition, it is not clear if the transition will remain in presence of disorder or if the randomness rounds it off  as in equilibrium systems \cite{AW}. 
%}
%
%
%{\it  \margincom{supprimer ?}
%Yet, one of the heuristic arguments put forward in \cite{tb} to justify the transition is the comparison with the so-called ``fully segregated model'', which is nothing but a periodic version of \cite{GKS}, that is microscopically very different from the disordered one.
%} 

\medskip



Renormalization is often key to analyze multi-scale phenomena in disordered systems; we refer to \cite{Sznitman, vares} for a general overview and to \cite{bt} for an approach related to ours. 
Our renormalization scheme  controls rigorously the multi-scale   slow bond picture described in the  paragraph above. 
%This scheme  can also be interpreted in terms of homogenization of scalar conservation laws \cite{bb}.
A major difficulty compared to the single slow bond is that as one moves to larger scales, the typical maximum current 
associated to a given scale and the maximum current associated to the rare slow regions occurring 
at the same scale converge 
 to the same value $r/4$ (with $r$ the minimal value of the jump rates).
 %The current limitation in both instances comes from large stretches of minimum rate $r$,
%which restrict the current to the maximum value $r/4$ of a homogeneous TASEP {\color{black}  with jump rate $r$}. 
%This gives rise to a renormalization picture which can fruitfully be interpreted in terms of homogenization \cite{bb}. 
Thus a delicate issue is to show that this small current  difference exceeds the typical order of fluctuations at each scale, so that the slow-bond picture remains  valid at all scales.
%
%Heuristically, this infinite hierarchy of slow-bond blockages \cite{fk, kru} is expected to induce 
%a dynamics of  coalescing shocks and antishocks  on mesoscopic scales depending on the disorder distribution.  
To quantify this difference, we rely on an assumption on the decay of the maximum current in a finite box \eqref{condition_h_2}.
This assumption is satisfied  under a condition on the tail of the rate distribution $Q$ near its minimum $r$
 (see Lemma \ref{lemma_assumption}). 
Heuristics suggest that this assumption should be always valid although we have not been able
so far to prove this conjecture. 





\medskip





We achieve our renormalization scheme by formulating the problem in wedge LPP framework with columnar disorder and exponential random variables.
In the LPP framework, the phase transition takes the form of a {\it pinning} transition for the optimal path \cite{khh}:
the path gets a better reward  from vertical portions along slow parts of the disorder.
The core of this approach is to obtain a recursion between mean passage times at two successive scales.
Like many shape theorems \cite{mar}, our results partially extends to LPP with more general distributions. 
% and reduce the problem to bounding our passage times by a family of ``reference'' passage times, that correspond to trapezoidal flux functions.






\medskip


The paper is organized as follows. In Section \ref{sec:results}, we set up the notation and state our main result.
In Section \ref{sec:lpp}, we formulate the problem in the last passage percolation framework and introduce the reference flux and the passage time functions. In Section \ref{sec:renorm}, we introduce the renormalization procedure and describe the main steps of the proof.
In Section \ref{subsec:outline}, we prove a recurrence which links the passage time bounds of two successive scales. This is the heart of the renormalization argument.
In Section \ref{sec:other_proofs}, we study this recurrence in detail and show that it propagates the bounds we need from one scale to another.
In Section \ref{sec: prop passage time}, we establish an important fluctuation estimate needed in Section \ref{subsec:outline}. Finally, the proofs of our main theorems are completed in Section \ref{sec:completion}.
%




\section{Notation and results}\label{sec:results}

\subsection{TASEP with site disorder}
%
Let $\N:=\{0,1,\ldots\}$ (resp. $\N^*:=\{1,2,\ldots\}$) be the set of nonnegative (resp. positive) integers.
The disorder is modeled by  $\alpha=(\alpha(x):\,x\in\Z)\in{\bf A}:=[0,1]^\Z$, a stationary ergodic sequence of positive bounded random variables. The precise distribution of $\alpha$ will be defined in 
Section \ref{sec: results}.
%The distribution of $\alpha$ is denoted by $\mathcal P$, and expectation with respect to this distribution by $\mathcal E$.  
For a given realization of $\alpha$, we consider the TASEP on $\Z$ with site disorder $\alpha$. 
The dynamics is defined as follows.
A site $x$ is occupied by at most one particle which may jump with rate $\alpha(x)$ to site $x+1$ if it is empty.
A particle configuration on $\Z$ is of the form $\eta=(\eta(x):\,x\in\Z)$, where for $x\in\Z$, 
$\eta(x)\in\{0,1\}$ is the number of particles at $x$. The state space is ${\bf X}:=\{0,1\}^\Z$. The generator of the process is given by
%
\be\label{generator}
%
L^\alpha f(\eta)=\sum_{x\in\Z}\alpha(x)\eta(x)[1-\eta(x+1)]\left[
%
f\left(\eta^{x,x+1}\right)-f(\eta)
%
\right],
%
\ee
%
where $\eta^{x,x+1}=\eta-\delta_x+\delta_{x+1}$ denotes the new configuration after a particle has jumped from $x$ to $x+1$. 
%
\medskip

\noindent
\textbf{Current and flux function.} 
%
%
%The macroscopic flux function  may a priori be defined in different ways, whose equivalence will be established below.\\ \\
%
%
%{\em First point of view.} A  definition of $f$ can be given as the limit of the microscopic current starting from macroscopically homogeneous states. 
%
%
The macroscopic flux function $f$ can be defined as follows.
We denote by $J^\alpha_x(t,\eta_0)$ the rightward current across site $x$ up to time $t$, that is
the number of jumps from $x$ to $x+1$ up to time $t$,  in the TASEP $(\eta_t^\alpha)_{t\geq 0}$ starting from initial state $\eta_0$, and evolving in environment $\alpha$.
%Alternatively, if $\sum_{x>0}\eta_0(x)<+\infty$, we also have
%
%$$
%J^\alpha_x(t,\eta)=\sum_{y>x}\eta_t^\alpha(y)-\sum_{y>x}\eta_0^\alpha(y)
%=h(t,x+1)-h(0,x+1)
%$$
%where $(\eta_t^\alpha)_{t\geq 0}$ denotes the process with generator \eqref{generator}.
%
For $\rho\in[0,1]$, let $\eta^\rho$ be an initial particle configuration with  asymptotic particle density $\rho$ in the following sense:
%
\be
\label{uniform_profile}
%
\lim_{n\to\infty} \frac{1}{n} \sum_{x=0}^n \eta^\rho(x)
=
\rho
=\lim_{n\to\infty} \frac{1}{n} \sum_{x=-n}^0 \eta^\rho(x).
%
\ee
%It is natural to set
%
%Then Theorem \ref{th_sep} and variational coupling (see also \cite{sep}) imply that, for a.e. realization of $\alpha$,
%
We then set
%
\be
\label{flux_current}
f(\rho) := \lim_{t\to\infty}\frac{1}{t}J^\alpha_x(t,\eta^\rho),
\ee
%
where the limit is understood in probability with respect to the law of the quenched process.
Other definitions of the flux and the proof of their equivalence with the above definition can be found in \cite{bb}.
%
%
%Equivalent  definitions of the flux can be
%stated in terms of the steady state expectation of the microscopic flux function 
%%
%\be\label{current_micro}
%%
%j^\alpha_x(\eta)=\alpha(x)\eta(x)[1-\eta(x+1)]
%%
%\ee
%%
%through one of the following formulas:
%%
%\begin{eqnarray}
%%
%\label{flux_torus}
%%
%f(\rho) & := & \lim_{N\to\infty,\atop 
%k/N\to\rho} \int j^\alpha_x(\eta)d\nu_{N,k}^\alpha(\eta), \\
%%
%f(\rho) & := &  \int j^\alpha_x(\eta)d\nu_\rho^\alpha(\eta)\label{flux_line},\quad\rho\in\mathcal R.
%%
%%
%\end{eqnarray}
%%
%In \eqref{flux_torus}, $\nu_{N,k}^\alpha$ is the steady state for a system of $k$ particles evolving 
%according to the dynamics \eqref{generator} restricted to a ring of size $N$ instead of $\Z$. In \eqref{flux_line} (see \cite{bgrs4} for details), $\nu_\rho^\alpha$ is the unique extremal invariant measure with mean density $\rho$ for the system on $\Z$, $\mathcal R\supset\{0,1\}$ is the (unknown) closed set of densities for which this measure exists, and $f$ is interpolated linearly outside $\mathcal R$. 
%We refer to 
%%Appendix \ref{app:equivalence} 
%\cite{bb} for the proof of equivalence of these three definitions in a more general framework. \\ \\
%
%


It is shown in \cite{sep} that $f$ is a concave function, see \eqref{flux_legendre} below. 
It was conjectured in \cite{tb} that for i.i.d. disorder, the flux function $f$ exhibits a flat segment, that is an interval $[\rho_c,1-\rho_c]$ (with $0\leq\rho_c<1/2$) on which $f$ is constant (see Figure \ref{fig: flux truncated}). The proof of \cite{sly} uses a comparison with a homogeneous rate $r$ TASEP.
We introduce a different approach, based on renormalization and homogenization ideas, viewing the disordered model as a perturbation of a homogenous rate $1$
TASEP. This yields not only an independent proof of the existence of a flat segment, but also optimal estimates when the density of defects is small enough. 
%Apart from concavity, little can be said about $f$. This is due to the fact that invariant measures for site-disordered TASEP are not explicit (in contrast with particle-disordered TASEP or equivalently site-disordered zero-range process, which has explicit product invariant measures). 
%%In the case of i.i.d. disorder, it is shown in \cite{sep} that $f$ is symmetric, i.e. $f(1-\rho)=f(\rho)$ for $\rho\in[0,1]$.
%%
%One of the key issues is whether $f$ is strictly concave or not. It is conjectured in the physics literature (see e.g. \cite{tb}) that for i.i.d. disorder, the flux function $f$ exhibits a plateau, that is an interval $[\rho_c,1-\rho_c]$ (with $0\leq\rho_c<1/2$) on which $f$ is constant (see figure \ref{fig: flux truncated}). 
%
%
%\medskip
%
%
%
\subsection{ The flux and flat segment for rare defects}
\label{sec: results}
%
%
%
From now on, we consider i.i.d. disorder such that the support of the distribution of $\alpha(x)$ is contained in $[r,1]$, where $r > 0$ is  the infimum of this support.
%{\color{black} 
Then the flux is bounded from above by $r/4$.
%}
%
%
%
\begin{proposition}
\label{prop_max}
%
The maximum value of the flux function is given by 
%
$$\max_{\rho\in[0,1]}f(\rho) = r/4.$$
%
\end{proposition}
%{\color{black} 
This result comes from the fact that the current of the disordered system is limited by atypical large 
stretches  with jump rates close to $r$. On these atypical regions, the system behaves as a homogeneous rate $r$ TASEP
which has maximum current $r/4$.
A detailed proof can be found in Appendix \ref{appendix_max}.
%}
%
%
For our main results, we formulate additional assumptions on the distribution\margincom{\bf CH} of the environment. 
We assume that the disorder is a perturbation of the homogeneous case with rate 1.
Let $0<r<R<1$ and $Q$ be a probability measure on $[r,R]$, such that $r$ is the infimum of the support of $Q$. 
Given $\varepsilon\in(0,1)$ a ``small'' parameter, 
we define the distribution of $\alpha(x)$ by
%
\be\label{law_disorder}
Q_\varepsilon = (1-\varepsilon) \delta_1 + \varepsilon Q .
\ee
The law of $\alpha = (\alpha(x), x \in \Z)$ is the product measure with marginal $Q_\varepsilon$ at each site
$$
\mathcal P_\varepsilon(d\alpha):=\bigotimes_{x\in\Z}Q_\varepsilon[d\alpha(x)].
$$
%
Expectation with respect to $\mathcal P_\varepsilon$ is denoted by $\mathcal E_\varepsilon$. 
We can interpret this by saying that each site is chosen independently at random to be, 
with probability $1-\varepsilon$, a ``fast'' site with normal rate $1$, or with probability $\varepsilon$ to be a ``defect'' with rate distribution $Q$ bounded above away from $1$ and $0$.
Thus $\varepsilon$ is the mean density of defects. 
For example if $Q = \delta_r$, then the defects are slow bonds with rate $r < 1$.\\ \\
%
Let us denote by $f_\varepsilon$ the flux function \eqref{flux_current} for the disorder distribution
%{\color{black} $Q_\varepsilon$}.
Our main results
%is the occurrence of a plateau in the flux for sufficiently dilute disorder. This result
hold under a general assumption \hh\mbox{ on} the disorder distribution $Q$ which will 
be stated and explained in the next subsection.
%{\color{black}
Concretely,  assumption \hh\mbox{ is} easily implied by the following simple tail assumption:
%}
\begin{lemma}\label{lemma_assumption}
%
Assumption \hh\mbox{ }holds if the following condition is satisfied:
%
\be
\label{tail_assumption}
\mbox{for some }\kappa>1, \qquad Q \big( [r,r+u) \big)=O(u^\kappa)\mbox{ as }u\to 0^+.
\ee
%
%
\end{lemma}
%
\noindent
%
We now define {\color{black} the edge of the flat segment as}
%
\be
\label{def_critical_density}
%
\rho_c(\varepsilon):=\inf\left\{\rho\in\left[0,\frac{1}{2}\right]: 
\qquad
f_\varepsilon  \equiv  \frac{r}{4}\mbox{ on }[\rho,1-\rho]\right\}.
%
\ee
%
It follows from Proposition \ref{prop_max} that $\rho_c(\varepsilon)\leq 1/2$. 
It is also known (see \cite{sep}) that $f_\varepsilon$ is symmetric with respect to $\rho=1/2$, i.e.
%
\be
\label{symmetry_f}
%
\forall \rho\in[0,1],\quad f_\varepsilon (1-\rho)= f_\varepsilon (\rho).
%
\ee
%
Therefore, \eqref{def_critical_density} is equivalent to saying that the flat segment of $f_\varepsilon$ is the interval 
$[\rho_c(\varepsilon),1-\rho_c(\varepsilon)]$.
%
 %However we first state it under a concrete 
%\textcolor{black}{lower tail assumption on $\alpha(x)$, which easily implies the general assumption \hh.  
%
%
%\begin{theorem}
%\label{th_plateau_tail}
%%
%Assume $Q$ satisfies the lower tail assumption 
%%
%\be
%\label{tail_assumption}
%\mbox{for some }\kappa>1, \qquad Q \big( (r,r+u) \big)=O(u^\kappa)\mbox{ as }u\to 0^+.
%\ee
%%
%Then:\\ \\
%%
%\textcolor{black}{
%%
%%
%%the flux $f_\varepsilon$ has a plateau with  value $r/4$.
%%
%(i) Let
%%
%\be\label{def_critical_density}
%%
%\rho_c(\varepsilon):=\inf\left\{\rho\in\left[0,\frac{1}{2}\right]:\,f_\varepsilon\equiv\frac{r}{4}\mbox{ on }[\rho,1-\rho]\right\}
%%
%\ee
%%
%Then there exists $\varepsilon_0>0$ such that, for every $\varepsilon<\varepsilon_0$, $\rho_c(\varepsilon)<\frac{1}{2}$. Besides,
%%
%\be\label{limit_plateau}
%\lim_{\varepsilon\to 0}\rho_c(\varepsilon)=\frac{1}{2}\left(1-\sqrt{1-r}\right)
%\ee
%%
%(ii) For every $\rho\in[0,1]$
%%
%\be\label{dilute_limit}
%%
%\lim_{\varepsilon\to 0}f_\varepsilon(\rho)=\min\left[\rho(1-\rho),\frac{r}{4}\right]=:f_0(\rho)
%%
%\ee
%%
%}
%%
%\end{theorem}
%
%
\begin{theorem}
\label{th_plateau}
Under assumption \hh, 
there exists $\varepsilon_0>0$ such that $\rho_c(\varepsilon)<\frac{1}{2}$ for every $\varepsilon<\varepsilon_0$. Furthermore, the size of the flat segment is explicit when $\varepsilon$ vanishes:
%
%
\be\label{limit_plateau}
\lim_{\varepsilon\to 0}\rho_c(\varepsilon)= \rho_c(0),
\ee
%
with
%
\be
\label{def_limit_plateau}
\rho_c(0):=\frac{1}{2}\left(1-\sqrt{1-r}\right).
\ee
\end{theorem}

\begin{remark}
%
It follows from \eqref{limit_plateau} that the limiting value of the length $1-2\rho_c(\varepsilon)$ of the flat segment is $\sqrt{1-r}$.
The result of \cite{sly} is that $\rho_c(\varepsilon)<1/2$ for any $\varepsilon\in(0,1)$, without requiring 
{\color{black} assumption \hh or} \eqref{tail_assumption}.
The proof of \cite{sly}  yields an upper bound on $\rho_c(\varepsilon)$ implying that 
%
%$$
%\limsup_{\varepsilon\to 0}\rho_c(\varepsilon)\leq\frac{1+r}{4}
%$$
%
%and thus 
the limiting length of the flat segment is at least $(1-r)/2$.
%
\end{remark}
%

%
\textcolor{black}{
The next theorem characterizes the dilute limit \cite{kru} of the whole} flux function.
Let
%
\be
\label{def_d0}
%
\forall \rho\in[0,1], 
\qquad
f_0(\rho) := \min\left[\rho(1-\rho),\frac{r}{4}\right].
%
\ee
%
\begin{theorem}
\label{th_dilute_limit}
%
Under assumption \hh, uniformly over $\rho\in[0,1]$, one has 
% 
\be
\label{dilute_limit}
%
\lim_{\varepsilon\to 0}f_\varepsilon(\rho)= f_0(\rho).
%
\ee
\end{theorem}
%
It is important to note that, although $\rho_c(0)$ is the lower bound of the flat segment of $f_0$, 
{\color{black} the convergence \eqref{limit_plateau} is not a direct consequence
%does {\it not} follow 
from \eqref{dilute_limit}.}
% because the size of the flat segment does not depend continuously on the flux function. 
Theorem \ref{th_dilute_limit} does not imply the existence of the flat segment for given $\varepsilon$ either. However, the proofs of \eqref{limit_plateau} and \eqref{dilute_limit} are closely intertwined and both follow from our renormalization approach.
%, of which they are two consequences of different natures.
%
%One can show that a soft tail for $Q$ is sufficient to imply Assumption \hh :
%%
%\begin{lemma}
%\label{lemma_tail}
%%
%Assume that for some $\alpha>2$, $Q((r,r+u))=O(u^\alpha)$ as $u\to 0^+$. Then Assumption \hh holds.
%%
%Then there exists $\varepsilon_0>0$ such that, for every $\varepsilon<\varepsilon_0$, $f_\varepsilon$ has a plateau with flux value $r/4$.
%%
%\end{lemma}
%
%{\em Proof.}
%%
%
%To prove Theorem \ref{th_plateau}, we first show that
%%
%\begin{proposition}
%\label{prop_max}
%%
%For every $\varepsilon>0$, $\max_{\rho\in[0,1]}f_\varepsilon(\rho) \leq r/4$
%%
%\end{proposition}
%%
%The proof of Proposition \ref{prop_max} follows essentially from the existence of arbitrarily large stretches of sites with
%rates arbitrarily close to $r$. See Appendix \ref{appendix_max} for details.
%
\subsection{A general assumption}
%
Let us now state  {\color{black} assumption {\hh} which is used in Theorems \ref{th_plateau} and \ref{th_dilute_limit}. 
For this we first define the maximal current in a finite domain.}

%need the following definition.

\medskip


%\noindent
%\textbf{Maximal current in a finite domain.} 
Let $B = [x_1,x_2]$ be an interval in $\Z$. In the 
following, $\alpha_B:=(\alpha(x):\ x\in B)$ denotes the environment restricted to $B$.
Consider the TASEP in $B$  with the following boundary dynamics: a particle enters at site $x_1$ with rate 
\textcolor{black}{$\alpha(x_1-1)$} if this site is empty and leaves from site $x_2$ with rate $\alpha(x_2)$ if this site is occupied. 
{\color{black}
Note that this process
%, although spatially restricted to $B$, 
depends 
%in fact on the restriction of 
on the disorder in the larger box}
\be
\label{def_bdiese}
B^\#:=[x_1-1,x_2]\cap\Z .
\ee
From now, 
%our notational choice will be to 
\textcolor{black}{we} index all related objects by $B^\#$ (the domain of the relevant disorder variables) rather
than $B$ (the domain where particles evolve).
%
The generator of this process is given by 
%
\begin{align}
%
L^\alpha_{B^\#} f(\eta) & :=  \sum_{x=x_1}^{x_2-1}\alpha(x)\eta(x)[1-\eta(x+1)]\left[
%
f\left(\eta^{x,x+1}\right)-f(\eta)
%
\right]\nonumber\\
%
& +  \alpha(x_1-1)[1-\eta(x_1)]\left[
%
f\left(\eta+\delta_{x_1}\right)-f(\eta)
%
\right]+\alpha(x_2)\eta(x_2)\left[
%
f\left(\eta-\delta_{x_2}\right)-f(\eta)
%
\right],
\label{generator_open}
%
\end{align}
%
where $\eta\pm\delta_x$ denotes the creation/annihilation of a particle at $x$.
We now define the maximal current relative {\color{black} to the process restricted} to $B$. 
%
%
\begin{definition}
\label{H_particle}
%
The maximal current $j_{\infty,B^\#}(\alpha_{B^\#})$ is the stationary current in the open system defined above, i.e. (independently of $x=x_1,\ldots,x_2-1$)\margincom{{\bf CH} ajoute numero eq}
%\margincom{\textcolor{black}{CH c'est bien $1$ devant la 1ere integrale du 2d membre}}
%
%
\begin{eqnarray}
\label{eq: max flux def}
j_{\infty,B^\#}(\alpha_{B^\#}) & = & \int\alpha(x)\eta(x)[1-\eta(x+1)]d\nu^\alpha_{B^\#}(\eta)\\
%
& = & 
\int 
\textcolor{black}{ \alpha(x_1-1)}
 [1-\eta(x_1)]d\nu^\alpha_{B\#}(\eta)=\int\alpha(x_2)\eta(x_2)d\nu^\alpha_{B^\#}(\eta) , 
\nonumber
\end{eqnarray}
%
where $\nu^\alpha_{B^\#}$ is the unique invariant measure for the process on $B$ with generator $L^\alpha_{B^\#}$.
%
\end{definition}
%
%
\begin{remark}
One can see that
the right-hand side of \eqref{eq: max flux def} is independent of $x$ 
by writing that the expectation under $\nu^\alpha_{B^\#}$ of $L^\alpha_{ B^\#}\eta(x)$ for $x\in[x_1,x_2]\cap\Z$ (which yields the difference of two consecutive integrals in \eqref{eq: max flux def}) is zero.
\end{remark}
%
%
To simplify notation, we shall at times omit the dependence on $\alpha_{B^\#}$ and write $j_{\infty,B^\#}$.
It is well-known \cite{der} that in the homogeneous case, i.e. when $\alpha(x) = r$ for all $x$
in $[0,N]$  (with $r$ a positive constant), then $j_{\infty,[0,N]}$ is no longer a random variable and
\be
\label{eq: courant homogene}
\lim_{N \to \infty} j_{\infty,[0,N]} = \inf_N j_{\infty,[0,N]}  = \frac{r}{4} \, . 
\ee
%
%In the next section, we will formulate an assumption on the statistics of the 
%maximal current $j_{\infty,B}(\alpha_B)$ when the size of the domain $B$ diverges.
%
%As the jump rates are larger or equal than $r$, we deduce from \eqref{eq: courant homogene} that the maximal current $j_{\infty,[0,N]}$ in large boxes cannot be smaller than $\frac{r}{4}$.
%
In fact, explicit computations \cite{der} show that, for some constant $C>0$,
%
\be\label{decay_r}
j_{\infty,[0,N]}\geq \frac{r}{4}+\frac{C}{N}.
\ee
%
The quantity $j_{\infty,[0,N]}(\alpha_{[0,N]})$ is a function of the environment which measures the speed of decay 
of the maximum current in a box to $r/4$ as the size of the box increases. 
%The homogeneous environment with value $r$ is a degenerate case of disorder for which the explicitely known flux function  $\rho\mapsto r\rho(1-\rho)$ has no flat segment. Therefore, a slower rate decay of $j_{\infty,[0,N]}(\alpha_{[0,N]})$ than in this extreme case means that we do have a certain amount of disorder.
{\color{black} Assumption {\hh}, stated below, requires that with high probability on the disorder the decay of the maximal current towards $r/4$ is slightly slower than \eqref{decay_r}.}
%that we now state for Theorems \ref{th_plateau} and \ref{th_dilute_limit} can be interpreted  
%by saying that any slower rate of decay of the maximal current than in \eqref{decay_r} is indeed sufficient. Thus this assumption is in some sense optimal.

\medskip

\noindent
{\bf Assumption \hh}.
%
{\it There exists $b\in(0,2)$, $a>0$, $c>0$ and $\beta>0$ such that, for $\varepsilon$ small enough, the following holds for any $N$}: 
%
\be
\label{condition_h_2}
%
\mathcal P_\varepsilon
\left(
j_{\infty,[0,N]} ( \alpha_{[0,N]}  )  
\leq\frac{r}{4}+\frac{a}{N^{b/2}} \right) 
\leq \frac{c}{N^\beta} \, .
%
\ee
Note that if assumption \hh\, is satisfied for some $b\in(0,1)$, it is satisfied {\it a fortiori} for $b=1$.
Thus, from now on, without loss of generality, we will assume that $b\in[1,2)$.
{\color{black}  We stress the fact that the condition $b<2$ is borderline as 
a simple comparison with the homogeneous case \eqref{decay_r} 
leads to a control of the decay for  $b=2$.}

\medskip


We have not been able to prove that assumption {\hh} is satisfied for Bernoulli disorder $Q=\delta_r$, although we believe this is true. However, as stated in  Lemma \ref{lemma_assumption}, the tail assumption \eqref{tail_assumption} implies {\hh}. 
%
%This is proved using the following lemma, which relies on a standard coupling argument.
% (see its proof in Appendix \ref{app_coupling}).
%
%
%\begin{lemma}
%\label{lemma_j_alpha}
%%
%Assume $\alpha$ and $\alpha'$ are two environments such that \textcolor{black}{$\alpha(x)\leq\alpha'(x)$ for all $x\in\Z$.} 
%Then, for every $N\in\N$, $j_{\infty,[0,N]}(\alpha_{[0,N]})\leq 
%j_{\infty,[0,N]}(\alpha'_{[0,N]})$.
%%
%\end{lemma}
%
%
\begin{proof}[Proof of Lemma \ref{lemma_assumption}]
%
Let $\alpha^\star:=\min_{x\in [0,N]}\alpha(x)$. 
It follows from 
{\color{black}  a standard coupling argument that the flux is monotonous with respect to the jump rates}
%Lemma \ref{lemma_j_alpha}  that
%
\be
\label{simple_coupling}
j_{\infty,[0,N]}(\alpha_{[0,N]}) \geq                                                            
j_{\infty,[0,N]} \big( \alpha^\star,\ldots,\alpha^\star \big),
%=\alpha^\star j_{\infty,[0,N]} \big(1,\ldots,1 \big).
\ee
where 
$j_{\infty,[0,N]} \big( \alpha^\star,\ldots,\alpha^\star \big)$ 
stands for the current of a homogeneous TASEP {in $[1,N]$ with bulk, exit and entrance rates
$\alpha^\star$.
%
By \eqref{decay_r}, it is larger than $\alpha^\star/4$, so that $j_{\infty,[0,N]}(\alpha_{[0,N]})  \geq\alpha^\star/4$.
Thus assumption \mbox{\hh} will be implied by control of $\alpha^\star$. 
Using  the tail of the distribution $Q$ \eqref{tail_assumption}, we get }
%
$$
\mathcal P_\varepsilon\left(
%
\min_{x \in[0,N]}\alpha(x)\leq r+\frac{a'}{N^{b/2}}
%
\right) \leq N Q \left( [r,  r+\frac{a'}{N^{b/2} } ) \right)     
\leq \frac{c'}{N^\beta},
$$
%
for some well chosen parameters $a'>0$ , $c'>0$, $b\in(0,2)$, $\beta>0$.
%
This follows from elementary computations.
%
\end{proof}
%
%
%\medskip
%
%For general distribution $Q$, we do not know how to check this assumption. However 
% if $Q$ has a soft enough tail near its minimum value $r$ one will prove in Lemma \ref{lemma_tail} that condition \hh holds.
%
%
\section{last passage percolation approach}\label{sec:lpp}
%
The derivation of Theorems \ref{th_plateau}  and \ref{th_dilute_limit} relies on a reformulation of the problem in terms of last passage percolation.
%
%
%
\subsection{Wedge last passage percolation} 
%
Let $Y=(Y_{i,j}:\,(i,j)\in\Z\times\N)$ 
%
%\textcolor{black}
%
{be} an i.i.d. family of exponential random variables with parameter 1 independent of the 
%
%\textcolor{black}
%
{environment} $(\alpha(i):\ i\in\Z)$. In the following, these variables will sometimes be called {\it service times}, in reference
to the queuing interpretation of TASEP.
%We set ${Y}'_{i,j}=Y_{i,j}/\alpha(i)\sim\mathcal E(\alpha(i))$. 
The distribution of $Y$ is denoted by $\Prob$ and {\color{black} the} expectation with respect to this distribution by $\Exp$.
%
Let 
$$
\mathcal W:=\{(i,j)\in\Z^2:\,j\geq 0,\,i+j\geq 0\} \, .
$$ 
Index $i$ represents a site and index $j$ a particle.
Given two points $(x,y)$ and $(x',y')$ in $\Z\times\N$, we denote by  $\Gamma((x,y),(x',y'))$ the set of paths
$\gamma=(x_k,y_k)_{k=0,\ldots,n}$ such that $(x_0,y_0)=(x,y)$, $(x_n,y_n)=(x',y')$, and $(x_{k+1}-x_k,y_{k+1}-y_k)\in\{(1,0),(-1,1)\}$ for every $k=0,\ldots,n-1$.
Note that  $\Gamma((x,y),(x',y'))=\emptyset$ if $(x'-x,y'-y)\not\in\mathcal W$. Given a path $\gamma\in\Gamma((x,y),(x',y'))$, its passage time is defined by
%
\be\label{def_passage}
T^\alpha(\gamma):=\sum_{k=0}^n \frac{Y_{x_k,y_k}}{\alpha(x_k)}.
\ee
%
The last passage time between $(x,y)$ and $(x',y')$ is defined by 
%
\be\label{def_last}
T^\alpha((x,y),(x',y')):=\max\{T^\alpha (\gamma):\,\gamma\in\Gamma((x,y),(x',y'))\}.
\ee
%
We shall simply write $T^\alpha(x,y)$ for $T^\alpha((0,0),(x,y))$. This quantity has the following particle interpretation.
%\eqref{def_etastar}.
%The initial particle at $x<0$ is labeled $-x-1$. 
%
%Start the TASEP with the initial
%configuration \eqref{def_etastar}.
%The initial particle at $x<0$ is labeled $-x-1$. 
%%
%Then $T^\alpha(i,j)$ is the time at which particle $j$ reaches site $i$.
%There is a correspondence between $T^\alpha(.,.)$ and the TASEP height profile 
%%
%$$H^\alpha(t,x):=\sum_{y\in\Z,\,y\geq x}\eta_t^\alpha(y)$$
%%
%given by
%%
%\begin{eqnarray*}
%%
%\label{corr_1} H^\alpha(t,x) & = & \min\{y\in\N:\,T^\alpha (x,y)>t\}\\
%%
%\label{corr_2} T^\alpha(x,y) & = & \inf\{t\in[0,+\infty):\,H^\alpha (t,x)> y\}
%%
%\end{eqnarray*}
%
For $(t,x)\in[0,+\infty)\times\Z$, let
%
\begin{eqnarray*}
%
%
H^\alpha(t,x) =  \min\{y\in\N: \quad T^\alpha (x,y)>t\}
\quad \text{and} \quad 
\eta_t^\alpha(x)  =  H^\alpha(t,x-1)-H^\alpha(t,x).
\end{eqnarray*}
%
Then $(\eta_t^\alpha)_{t\geq 0}$ is a TASEP with generator \eqref{generator} and initial configuration 
\textcolor{black}{$\eta^*=\indicator{\Z\cap(-\infty,0]}$}, and 
$H^\alpha$ is its height process.
%
Besides, if we label particles initially so that  the particle at $x\leq 0$ has label $-x$, 
then for $(x,y)\in\mathcal W$, $T^\alpha(x,y)$ is the time at which particle $y$ reaches site  $x+1$.
Let us recall the following result from \cite{sep}.
%
\begin{theorem}
\label{th_sep}
%
Let $\mathcal W':=\{(x,y)\in\R^2:\,y \geq 0,\,\,x+y\geq 0\}$.
% 
For {\color{black} $\mathcal P$-a.s.} realization of the disorder $\alpha$, the function\margincom{\textcolor{black}{j'aurais mis a.e. ?}}
%
\be
\label{def_limtime}
(x,y)\in\mathcal W'
\mapsto
\tau(x,y):=\lim_{N\to\infty}  \frac{1}{N}  T^\alpha([Nx],[Ny])
\ee
%
is well-defined in the sense of a.s. convergence with respect to the distribution of $Y$. 
It is finite, positively $1$-homogeneous and superadditive (thus concave). The function
%
\be
\label{def_limheight}
(t,x)\in[0,+\infty)\times\R\mapsto h(t,x):=\lim_{N\to\infty}\frac{1}{N}H^\alpha([Nt],[Nx])
%= t k \left( \frac{x}{t} \right) 
\ee
is well-defined in the sense of a.s. convergence with respect to the distribution of $Y$. It is finite, positively $1$-homogeneous and subadditive (thus convex). These functions do not depend on $\alpha$ and are related through
%
\begin{eqnarray}
%
\label{corr_3} h(t,x) & = & \inf\{y  \in [0,+\infty): \quad  \tau(x,y)>t\},\\
%
\label{corr_4} \tau(x,y) & = & \inf\{t  \in[0,+\infty): \quad h(t,x)\geq y\}.
%
\end{eqnarray}
%
\end{theorem}
%
By homogeneity, the function $h$ in \eqref{def_limheight} is of the form
%
\be\label{def_limheight_k}
h(t,x)=t k \left( \frac{x}{t} \right) 
\ee
%
for some convex function $k:\R\to\R^+$. 
%
It is known that for homogeneous TASEP (that is $\alpha(x)=1$ for all $x$), we have
\begin{eqnarray}
\label{eq: homogeneous LPP}
\tau(x,y)=(\sqrt{x+y}+\sqrt{y})^2 ,\quad 
%h(t,x)=t\,k(x/t),\quad\mbox{ with}&&\\
k(v)=\frac{(1-v)^2}{4}\indicator{[-1,1]}(v)-v\indicator{(-\infty,-1)}(v) .\nonumber
\end{eqnarray}





\subsection{Reformulation of Theorems \ref{th_plateau}   and \ref{th_dilute_limit}}

In this section, we are going to rewrite the flux in the last passage framework and show that Theorem \ref{th_plateau} can be deduced from a statement on the passage time.
It is shown in \cite{sep} that the macroscopic flux function $f$ is related to $k$ (defined in \eqref{def_limheight_k}) by the convex duality relation
%
\be
\label{flux_legendre}
f(\rho):=\inf_{v\in\R}[k(v)+v\rho],\quad\rho\in[0,1]
\ee
%
which implies concavity of $f$. 
%
%
We now
introduce a family of ``reference'' macroscopic flux functions and associated macroscopic passage time and height functions. Let $0\leq\rho_c\leq1/2$ and $J \geq 0$.
For 
%$t>0$, $x\in\R$, $y\geq(-x)^+$ and 
$\rho\in[0,1]$, we define (see Figure \ref{fig: flux truncated})
%
%
%We define $f^{\rho_c,J}(\rho)$ (where $\rho\in[0,1])$ and $\tau^{\rho_c,J}(x,y)$ (where $(x,y)\in\R\times[0,+\infty)$) by
%
\be
%
%
\label{ref_flux}
%
f^{\rho_c,J}(\rho)  : = 
%
%\frac{J}{\rho_c}\rho\indicator{[0,\rho_c)}(x)+
%J\indicator{[\rho_c,1-\rho_c)}(x)+\frac{J}{\rho_c}(1-\rho)\indicator{[1-\rho_c,1]}(x)\\
%
J\min\left(
%
\frac{\rho}{\rho_c},\frac{1-\rho}{\rho_c},1
%
\right).
%
\ee



\begin{figure}[htpb]
  \centering
%\includegraphics[width=.4\columnwidth]{truncated_flux.pdf}
\includegraphics[width=.4\columnwidth]{truncated_flux.pdf}
\caption{\tiny 
{\color{black}
The homogeneous TASEP flux $f(\rho) = \rho (1-\rho)$ is represented in dotted line and 3 graphs of modified fluxes $f^{\rho_n,J_n}$ are depicted in plain line.
The renormalization strategy   is to bound from below the flux at the scale $n$ by $f^{\rho_n,J_n}$ and to use this information to control the lower bound on the flux at the scale $n+1$. As depicted in the figure, the fluxes $J_n$ decay to  $r/4$ when the scale grows. The width of the flat segment $[\rho_n, 1- \rho_n]$ is also shrinking at each step but remains controlled. 
When the dilution $\varepsilon$ tends to 0, the limiting flux $f_0$ defined in \eqref{def_d0} is the flux $f(\rho)$ truncated in 
$[\rho_c(0), 1- \rho_c(0)]$ at the level $r/4$ (dashed line). 
For $\varepsilon$ small enough, $J_1$ can be chosen very close to $r/4$ and $\rho_1$ close to $\rho_c(0)$. Furthermore for small $\varepsilon$, 
the flat segment $[\rho_n, 1- \rho_n]$ is almost unchanged at each scale and
this leads to the convergence in Theorem \ref{th_plateau}.}
 } 
  \label{fig: flux truncated} 
\end{figure}



Given Proposition \ref{prop_max}, 
{\color{black} the occurence of a flat segment in Theorem \ref{th_plateau} boils down to proving 
the existence of $\varepsilon_0>0$ and  $\rho \in[0,1/2)$ such 
that the flux remains above $r/4$  for densities in  $[\rho, 1- \rho]$:}
%
\be
\label{lower_bound_0}
\forall \varepsilon < \varepsilon_0, \qquad 
f_\varepsilon  \geq f^{{\rho},r/4} .
%\quad \text{with} \quad
%\rho \in [0,1]
%
\ee
%
{\color{black}
As $f_\varepsilon \leq r/4$ by Proposition \ref{prop_max}, lower bound \eqref{lower_bound_0} implies that $f_\varepsilon$ equals $r/4$ on $[\rho, 1- \rho]$.
Since $f_\varepsilon$ is concave and symmetric \eqref{symmetry_f}, $\rho_c(\varepsilon)$ in \eqref{def_critical_density} is characterized by:}
%Moreover, since $f_\varepsilon$ is a concave function such that $f_\varepsilon(0)=f_\varepsilon(1)=0$, has maximum value $r/4$ by Proposition \ref{prop_max},  and satisfies \eqref{symmetry_f}, we can characterize $\rho_c(\varepsilon)$ in \eqref{def_critical_density} as:
%
\be
\label{charac_critical}
%
\rho_c(\varepsilon)=\inf\left\{
%
\rho\in[0,1/2]: \quad f\geq f^{\rho,r/4}
%
\right\}.
%
\ee
%
%
%The sufficient condition \eqref{lower_bound_0} for the \textcolor{black}{flat segment} is actually necessary since, by Theorem \ref{th_sep} and \eqref{flux_legendre}--\eqref{flux_current}, 
 %$f_\varepsilon$ is a concave function such that $f_\varepsilon(0)=f_\varepsilon(1)=0$. 
%
The convex conjugate of  $f^{\rho_c,J}$ through Legendre duality \eqref{flux_legendre} is defined for $x\in\R$ by
%
\begin{eqnarray}
%
\label{ref_height_2}
%
k^{\rho_c,J}(x) & := & (-x)\indicator{(-\infty,-J/\rho_c)}(x)\\
%
\nonumber & + & [J-(1-\rho_c)x]\indicator{[-J/\rho_c,0)}(x)
+[J-\rho_c x]\indicator{[0,J/\rho_c)}(x).
%
\end{eqnarray}
%
Finally, one can associate to $k^{\rho_c,J}$ a  passage time function and a height function, related by \eqref{corr_3}--\eqref{corr_4},  and defined for $x\in\R$ and $y\geq x^-$
by
%
\begin{eqnarray}
\label{ref_time} 
\tau^{\rho_c,J}(x,y) := \frac{\rho_c x^+ - (1-\rho_c)x^-+y}{J} 
%
%\label{ref_height}
\quad \text{and} \quad 
h^{\rho_c,J}(t,x)   :=  t k^{\rho_c,J}(x/t) ,
%
%
\end{eqnarray}
where $x^+ = \max \{ x,0 \}$ and $x^- = -\min \{ x,0 \}$.
It follows from \eqref{corr_3}, \eqref{def_limheight_k} and \eqref{flux_legendre} that
%
\be
\label{inverse_order}
%
f\geq f^{\rho,J} \quad \Leftrightarrow  \quad \tau\leq\tau^{\rho,J}.
%
\ee
%
Hence, the quantity $\rho_c(\varepsilon)$ in \eqref{def_critical_density} can be defined equivalently as follows:
\be
\label{critical_density_2}
%
{\rho}_c(\varepsilon)=\inf\{\rho\in[0,1/2]:
\quad \tau_\varepsilon  \leq  \tau^{\rho,r/4}\}.
%
\ee
%
Thus the lower bound \eqref{lower_bound_0} on the flux  can be rephrased in terms of an upper bound on the last passage time.
%
%\textcolor{black}{In view of \eqref{corr_3}, \eqref{def_limheight_k} and \eqref{flux_legendre}}, 
Theorems \ref{th_plateau}--\ref{th_dilute_limit} are consequences of the following theorems, which will be proved in the next sections.
%
%
\begin{theorem}
\label{thm: passage upper}
Let $\tau_\varepsilon$ be the limiting passage time defined by \eqref{def_limtime} when the environment has distribution $\mathcal P_\varepsilon$. Then, under assumption \hh, 
%
there exist $\varepsilon_0>0$ and $\rho< 1/2$ such that 
\be
\label{lower_bounds}
\textcolor{black}{\forall \varepsilon < \varepsilon_0},
\qquad 
\tau_\varepsilon \leq \tau^{\rho,r/4},
\ee
with $ \tau^{\rho,r/4}$ defined in \eqref{ref_time}.
In particular, $\tau_\varepsilon(.,y)$ has a cusp at $x=0$ 
\textcolor{black}{
and
the optimal value $ {\rho}_c(\varepsilon)$ introduced in \eqref{critical_density_2} converges in the dilute limit:}
%
\be
\label{limit_rho'}
\lim_{\varepsilon\to 0} {\rho}_c(\varepsilon)=\frac{1}{2}(1-\sqrt{1-r}).
\ee
\end{theorem}
%
\begin{theorem}
\label{th:passage_dilute}
%
\textcolor{black}{The  passage time function $\tau_\varepsilon$ converges in the dilute limit:}
%
\textcolor{black}{
%
\be
\label{def_tau0}
%
\lim_{\varepsilon\to 0}\tau_\varepsilon(x,y)=\tau_0(x,y):=\left\{
%
\ba{lll}
%
%y\frac{1-2\rho_c(0)}{\rho_c(0)^2}\\ \\
%
(\sqrt{x+y}+\sqrt{y})^2 &  \mbox{if} & y\leq x^+ y_1^1(0)-x^-y_1^{-1}(0)\\ \\
%x\not\in \left(-\frac{y}{y_1^{-1}(0)},\frac{y}{y_1^{1}(0)}\right)\\
%
\tau^{\rho_c(0),r/4}(x,y) & \mbox{if} & y> x^+ y_1^1(0)-x^-y_1^{-1}(0)
%x\in\left(-\frac{y}{y_1^{-1}(0)},\frac{y}{y_1^{1}(0)}\right) 
%
%
\ea
%
\right.
%
\ee
%
}
%
\textcolor{black}{where $\tau_0$ is the counterpart of the flux function $f_0$
defined in \eqref{dilute_limit} and}
\margincom{{\color{red} Il y a 2 valeurs de $y_1(0)$ selon $\sigma = \pm 1$ ?\\
et aussi proposition 4.2 ?
}\textcolor{black}{en effet, j'ai note $y_1^\sigma$, cf. preuve prop 4.5}}
%
\be
\label{def_y1}
%
y_1^{\textcolor{black}{1}}(0):=\frac{\rho_c(0)^2}{1-2\rho_c(0)}\in[0,+\infty],\quad
\textcolor{black}{
%
y_1^{-1}(0):=\frac{[1-\rho_c(0)]^2}{1-2\rho_c(0)}\in[0,+\infty]
%
}
%
\ee
where $\rho_c(0)$ was introduced in \eqref{def_limit_plateau}.
%
\end{theorem}
 
 
Theorem \ref{thm: passage upper} can be partially extended to LPP with general service-time distribution and heavier tails.
In this case the particle interpretation is less standard, though the process can be viewed as a non-markovian TASEP (see e.g. \cite{khg}  and \cite{sep}).
Our approach (and the extension just explained) also applies to other LPP models with columnar disorder (in the wedge picture) or diagonal disorder (in the square picture), like for instance the $K$-exclusion process \cite{sep}.






\subsection{Last passage reformulation of assumption \hh}



We  will  reformulate condition \hh\mbox{ in }the last passage setting. To this end, we define restricted passage times. Let $B=[x_1,x_2]\cap\Z$ (where $x_1,x_2\in\Z$) be a finite interval of $\Z$. If $(x,y)$ and $(x',y')$ are such that $x$ and $x'$ lie in $B$, we define $\Gamma_B((x,y),(x',y'))$ as the subset of $\Gamma((x,y),(x',y'))$ consisting of paths
$\gamma$ that lie entirely inside $B$ in the sense that $x_k\in B$ for every $k=0,\ldots,n$. We then define 
%
\be\label{restricted_time}
T^\alpha_B((x,y),(x',y')):=\max \Big\{T^\alpha (\gamma):\,\gamma\in\Gamma_B((x,y),(x',y')) \Big\}.
\ee
%
%and again write $T^\alpha_B(x,y)$ for \textcolor{black}{$T^\alpha_B((x_1,0),(x,y))$}. 
The counterpart of Definition \ref{H_particle} is
%
\begin{lemma}
\label{lemma_reform}
Let $B=[x_1,x_2]\cap\Z$.
The limit
%
\be\label{def_mintime}
T_{\infty,B}(\alpha_B):=\lim_{m\to\infty} 
%
%\textcolor{black}
%
{\frac{1}{m}}  T^\alpha_{B}((x_0,0),(x_0,m))
=\sup_{m\in\N^*}   
%
%\textcolor{black}
%
\Exp\left[{\frac{1}{m}}  T^\alpha_{B}((x_0,0),(x_0,m))\right]
%
\ee
%
exists $\Prob$-a.s. for $x_0\in B$, does not depend on the choice of $x_0$, and defines a random
variable depending only on the disorder restricted to $B$.
Besides, we have
%
\be\label{max_current_lpp}
T_{\infty,B}(\alpha_B)= \frac{1}{j_{\infty,{B}}(\alpha_{B})}  ,
\ee
%
where 
%%
$j_{\infty,B}(\alpha_{B})$ is the stationary current \eqref{eq: max flux def} in the open system restricted to 
%
\be
\label{def_bprime}
B':=[x_1+1,x_2]\cap\Z.
\ee
%
%as in Definition \ref{H_particle}.
%
\end{lemma}
%
\noindent
%
\textcolor{black}{
%\begin{remark}
Note that in \eqref{eq: max flux def},  $j_{\infty,B^\#}(\alpha_{B^\#})$ was defined
as the maximum current for the TASEP in $B$.  
By \eqref{def_bdiese} and \eqref{def_bprime},
$(B')^\#=B$ so that  the above lemma is  consistent with \eqref{eq: max flux def}.
%\end{remark}
}
The proof of  Lemma \ref{lemma_reform} is postponed to Appendix \ref{appendix_lpp}.


\medskip

To simplify notation, we shall at times omit $\alpha_B$ and write $T_{\infty,B}$,  $j_{\infty,B}$.
We can now restate  condition \hh\, {\color{black} in terms of last passage time}:

\smallskip

\noindent
{\bf Assumption \h}. {\it There exists $b\in(0,2)$, $a>0$, $c>0$ and $\beta>0$ such that, for $\varepsilon$ small enough, one has for any $N\in\N^*$:}
%
\be
\label{condition_h}
%
\mathcal P_\varepsilon\left(
%
T_{\infty,[0,N]}  ( \alpha_{[0,N]} ) 
\geq\frac{4}{r}-\frac{a}{N^{b/2}}
%
\right)\leq \frac{c}{N^\beta} \, .
%
\ee
%
The constants $a,c$ in \eqref{condition_h_2} are different from those in \eqref{condition_h}, but $b$ and $\beta$ are the same.




 
\section{Renormalization scheme}\label{sec:renorm}


From now, we are going to focus on the last passage percolation model in order to prove Theorems  \ref{thm: passage upper} and \ref{th:passage_dilute}.
We first describe a renormalization procedure to show that a bound of the form \eqref{lower_bounds}
holds with high probability at every scale (see Proposition \ref{prop_block}  below).



%In order to prove \eqref{lower_bounds}, we use a renormalization procedure. The idea is that a bound of the form \eqref{lower_bounds}
%holds with high probability at every scales (see Proposition \ref{prop_block}).
%


\subsection{Definition of blocks}
%
Let $n\in\N^*$ be the renormalization ``level'' and $K_n=K_n(\varepsilon)$ the size of a renormalized block of level $n$ (by block we mean a finite subinterval of $\Z$). For $n=1$, we initialize $K_1=K_1(\varepsilon)$ and define a block $B$ of order $1$ to be good if it contains no defect, i.e. $\alpha(x)=1$ for every $x\in B$. Otherwise, the block is said to be bad.

\medskip

%
For $n\geq 1$, we set $K_{n+1}=l_n K_n$, where $l_n= \lfloor K_n^\gamma \rfloor$ with $\gamma\in(0,1)$. For $n\geq 1$, a block $B_{n+1}$ of order $n+1$ has size $K_{n+1}$ and is partitioned  into $l_n$ disjoint blocks of level $n$. This block is called ``good''
if it contains at most one bad subblock of level $n$, and if condition \eqref{condition_good} below  on the maximum current in the block holds:
%
\be
\label{condition_good} 
j_{\infty,B_{n+1}} 
\geq j_{n+1}
\quad \text{with} \quad 
j_{n+1} : =  \frac{r}{4}+\frac{a}{K_{n+1}^{b/2}} ,
\ee
where the constants $a,b$ were defined in 
%\eqref{condition_h} and 
\textcolor{black}{\eqref{condition_h_2}}.
Otherwise $B_{n+1}$ is said to be bad. We stress the fact that the status (good or bad) of $B_{n+1}$ depends only on the disorder variables $\alpha_{B_{n+1}}$ in $B_{n+1}$ and not on the exponential times $Y_{i,j}$.

\medskip

The renormalization is built such that large blocks are good with high probability.
Let $q_n(\varepsilon)$ denote the probability under $\mathcal P_\varepsilon$ that 
the block $[0,K_{n}-1]\cap\Z$, at level $n$, is bad.
%
\begin{lemma}
\label{lemma_good}
%
Suppose that assumption \hh\,  holds and set 
$$
K^*(\varepsilon):=\left( \frac{2c}{\varepsilon} \right)^{\frac{1}{\beta+1}}, 
\quad
%
K_*:= 2+ (4c)^{ \frac{1}{\beta-\gamma(\beta+2)} },
$$
%
%
\begin{equation}
\label{eq: parameters}
\gamma_0:=\frac{\beta}{\beta+2}, 
\quad
\varepsilon_0:=\min\left\{1,2^{-\beta}c,
(2c)\left[3+(4c)^{-\frac{1}{\beta-\gamma(\beta+2)}} \right]^{\beta+1}
\right\},
\end{equation}
with the constants $c,\beta$ appearing in  \eqref{condition_h_2} and \eqref{condition_h}.
Then 
%there exists $\gamma_0, \varepsilon_0 >0$ such that 
for all  $\gamma\in(0,\gamma_0)$ and $\varepsilon \leq \varepsilon_0$,  
{\color{black} there is an integer $K_1 (\varepsilon)$ in}
the interval $[K_*, K^*(\varepsilon)]$ such that 
%contains at least one integer, and if $K_1$ is any such integer,
%the following holds:
%$K^*(\varepsilon)\geq\max(1,K_*)$, and
%for every $K_1 \in [\textcolor{black}{\max(2,K_*)}, K^*(\varepsilon)]$:
%
\be
\label{conclusion_lemma_good}
\forall \varepsilon < \varepsilon_0, \qquad \lim_{n\to\infty}q_n(\varepsilon)=0 
\quad \text{and furthermore} \quad 
%\label{choice_K1}
%
\lim_{\varepsilon\to 0}K_1(\varepsilon)=+\infty.
\ee 
%
\end{lemma}
%
\begin{proof} 
%
For $n\geq 1$, let $\zeta_{n}=\frac{c}{K_{n}^{\beta}}$ be the upper bound in \eqref{condition_h}. 
Then, by definition of good blocks and independence of the environment, 
one obtains the recursive inequality
%
\begin{eqnarray*}
%
q_1 & \leq & K_1\varepsilon , \\
%
q_{n+1} & \leq & (l_n q_n)^2 + \zeta_{n+1},
\quad n\geq 1.
\end{eqnarray*}
%
Note that if for some $n$, we have $q_n\leq 2 \zeta_n$ and $\zeta_{n+1}\geq 4 l_n^2 \zeta_n^2$, then $q_{n+1}\leq 2 \zeta_{n+1}$.
Thus if we have $q_1\leq 2\zeta_1$ and $\zeta_{n+1}\geq 4 l_n^2 \zeta_n^2$ 
for all $n\geq 1$, then $q_n\leq 2 \zeta_n$ 
for all $n\geq 1$, implying $q_n(\varepsilon)\to 0$ as $n\to\infty$  provided $K_1\geq 2$.


On the one hand, $q_1\leq 2 \zeta_1$ follows from {\color{black} $K_1\leq K^*(\varepsilon)$.}
%
%\be
%\label{condition_1}
%K_1\leq K^*(\varepsilon).
%\ee
%
On the other hand, 
$\zeta_{n+1}\geq 4 l_n^2 \zeta_n^2$ is equivalent to
%
$$
K_n^{\beta-(\beta+2)\gamma}\geq 4 c,\quad\forall n\geq 1.
$$
%
Assuming $0\leq\gamma<\frac{\beta}{\beta+2}$, 
since $K_n$ is increasing in $n$,  the above inequality holds for all $n\geq 1$ if it holds for $n=1$, which is equivalent to 
\be
\label{condition_2}
K_1\geq K':= (4c)^{\frac{1}{\beta-\gamma(\beta+2)}}.
\ee
%
Finally,  setting $K_*:=2+K'$, we have $1+K_* \leq K^*(\varepsilon)$  if  $\varepsilon \leq \varepsilon_0$.
{\color{black}
Thus \eqref{conclusion_lemma_good} is satisfied by choosing the sequence $K_1(\varepsilon):=\lfloor K^*(\varepsilon)\rfloor$. }
%
%\be
%\label{choice_K1_comp}
%=
%%
%\left\lfloor
%\left(
%%
%\frac{2c}{\varepsilon}
%%
%\right)^{\frac{1}{\beta+1}}\right\rfloor,
%%
%\ee
%
%
%where $K^*(\varepsilon)$ was defined in Lemma \ref{lemma_good}.
%
\end{proof}
%



\subsection{Mean passage time in a block}
%
{\color{black} Note that the scale $K_n$ at level $n$ depends on $\varepsilon$ through $K_1(\varepsilon)$ 
(see \eqref{conclusion_lemma_good}) but we omit it in the notation for simplicity.}

The strategy to prove Theorem \ref{th_plateau} is now as follows. 
To each block 
$B=[x_0, x_1 := x_0+K_n-1]\cap\Z$ of  level $n$, we associate finite-size macroscopic restricted passage time functions (in the left and right directions) taking as origin either extremity of the block:
%
\begin{eqnarray}
\label{eq: path direction}
\begin{cases}
\tau^\alpha_{n,B}(1,y)  =  \Exp \left( \frac{1}{K_n} T_{B}^\alpha((x_0,0),(x_1, \lfloor K_n y\rfloor)) \right)
,\quad y\geq 0, \\
%
\tau^\alpha_{n,B}(-1,y) = \Exp \left( \frac{1}{K_n} T_{B}^\alpha((x_1,0),(x_0,\lfloor K_n y\rfloor)) \right)
,\quad y\geq 1,
\end{cases}
%
\end{eqnarray}
%
which depend only on the disorder $\alpha_{B_n}$.
To keep compact notation, we will write both functions in the form 
$\tau^\alpha_{B_n}(\sigma,y)$ with $\sigma = \pm 1$ and $y \geq \sigma^- = - \min\{\sigma,0\}$.


\medskip

The main step towards Theorem \ref{thm: passage upper}, stated in Proposition \ref{prop_block} below, is to prove that the mean passage time at each level $n$ remains bounded by 
the reference function \eqref{ref_time} with parameters $\rho_n,J_n$ appropriately controlled to ensure that the flat segment is preserved at each order.
\begin{proposition}
\label{prop_block}
%
For small enough $\varepsilon$, there exist sequences 
$(\rho_n=\rho_n(\varepsilon))_{n \geq 1}\in[0,1]^{ \N^*}$ 
and $(J_n=J_n(\varepsilon))_{n \geq 1}\in [0,+\infty)^{ \N^*}$
% and functions $(\sigma,y)\mapsto g_n(\sigma,y)$ defined for $\sigma\in\{-1,1\}$ and $y\geq\sigma^-$, 
% satisfying \eqref{nonparbound}, and 
such that: 

\smallskip

\noindent
(i) {\color{black} Uniformly over 
%the disorder in a 
good blocks $B$ at level $n$ and for every $\sigma\in\{-1,1\}$}\margincom{\textcolor{black}{$\alpha$ est fixe et B varie, donc plutot over good block ?}}
%, and every $y\geq \sigma^-$
%
\be
\label{bound_on_good_blocks}
\forall y\geq \sigma^-, \qquad 
{\color{black} \sup_{\text{good} \ B}} 
%\alpha_B }} \;
\tau^\alpha_{n,B}(\sigma,y) \leq %\textcolor{black}{g_n(\sigma,y)} \leq
\tau^{\rho_n,J_n}(\sigma,y),
\ee
%
(ii) $\lim_{n\to\infty} J_n=r/4$ and $J_n>r/4$ for all $n\in\N^*$, \\
%
(iii) $\limsup_{n\to\infty}\rho_n<1/2$ \textcolor{black}{
%
and with the definition \eqref{def_limit_plateau} of $\rho_c(0)$
$$
\limsup_{\varepsilon\to 0}\limsup_{n\to+\infty}
\rho_n(\varepsilon)\leq \rho_c(0)= \frac{1}{2}(1-\sqrt{1-r}).
$$
%
}
%
%(iv) $\lim_{n\to\infty}c_n=0$.
%
%\sc{+vanishing logarithmic corrections ?}
%
\end{proposition}
Once Proposition \ref{prop_block} is established, completing the proof of Theorem \ref{thm: passage upper} (and thus Theorem \ref{th_plateau}) is a relatively simple task,
which boils down to obtain a similar bound on {\it unrestricted} passage times (see  Section \ref{sec:completion}). 
\textcolor{black}{The upper bound $\tau^{\rho_n,J_n}$ is the counterpart, in the last passage percolation setting, of the modified flux $f^{\rho_n,J_n}$  depicted Figure \ref{fig: flux truncated}.}


{\color{black} 
The  derivation of Theorem \ref{th_dilute_limit} relies on  a refined version of \eqref{bound_on_good_blocks}
in the dilute limit.}
%that the functions $g_n$ in \eqref{nonparbound} enjoy the following property, where we write $g_{n,\varepsilon}$ instead of $g_n$ to emphasize dependence on $\varepsilon$.
%
\begin{proposition}\label{prop_dilute_limit}
%
For every $\sigma\in\{-1,1\}$ and \textcolor{black}{$\sigma^-\leq y<\sigma^+y_1^1(0)-\sigma^-y_1^{-1}(0)$}
%
\be\label{dilute_g}
%
\limsup_{\varepsilon\to 0}\limsup_{n\to+\infty}
{\color{black} \sup_{\text{good} \ B }} \;
\tau^\alpha_{n,B}(\sigma,y) 
%g_{n,\varepsilon}(\sigma,y)
\leq(\sqrt{\sigma+y}+\sqrt{y})^2.
\ee
\margincom{\textcolor{black}{idem}} where  $y_1(0)$ was defined in \eqref{def_y1}.
\end{proposition}
%
%








\subsection{Coarse-graining and recursion}
\label{subsec:coarse}


%We now define appropriate functions $g_n$ and sequences $(\rho_n)_{n\geq 1}$ and $(J_n)_{n\geq 1}$
%satisfying Propositions \ref{prop_block} and \ref{prop_dilute_limit}.

The strategy of the proof of Propositions \ref{prop_block} and \ref{prop_dilute_limit} is based on a coarse-graining procedure.
\textcolor{black}{
We will first show a general estimate for any good block $B$ at level $n \geq 1$ and $\sigma\in\{-1,1\}$\margincom{\textcolor{black}{idem}}\textcolor{black}{
\be
\label{nonparbound}
\forall y\geq\sigma^-, \qquad 
\sup_{\text{good} \ B}\;
%\alpha_B } \;
 \tau^\alpha_{n,B}(\sigma,y)\leq g_n(\sigma,y),
\ee
%
}
%
where $g_n(\sigma, \cdot)$ is a sequence of concave functions defined recursively in Proposition \ref{prop:induction_gen}. Then Propositions \ref{prop_block} and \ref{prop_dilute_limit} will be deduced from \eqref{nonparbound}.
%
}
%
\begin{proposition}
\label{prop:induction_gen}
%
{\color{black} Fix $C$ a large enough constant and set}
\be
\label{eq: delta n}
%$$
j_{n+1}:=\frac{r}{4}+\frac{a}{K_{n+1}^{b/2}}
,\quad l_n := \lfloor K_n^\gamma \rfloor \quad 
\mbox{ and }\quad
\delta_n:= C\frac{(\log K_{n+1})^{3/2}}{\sqrt{K_n}},
\ee
%
%for some constant $C>0$. 
%where $C$ is a large enough constant fixed once and for all. Then
%
the sequence $(g_n)_{n \geq 1}$ defined on $[\sigma^-,+\infty)$ by 
%
\margincom{CH 31/10 nouveau $g_1$ de la limite dilu\'ee}
%
\be\label{level_1}
%g_1=\tau^{\rho_1,J_1},\quad\mbox{with } \quad J_1\in(r/4,1/4),
%\quad 
%\rho_1=2J_1,
g_1(\sigma,y):=(\sqrt{\sigma+y}+\sqrt{y})^2
\ee
and 
\begin{align}
g_{n+1}(\sigma,y)  :=   \sup_{ \sigma^- \leq\bar{y}\leq \frac{l_n}{(l_n-1)} y}
\left\{
\left(1-\frac{1}{l_n}\right)\left[
%
g_n(\sigma,\bar{y})- \frac{\bar{y}}{j_{n+1}}
%
\right] \right\} 
+ \frac{y}{j_{n+1}}+  \frac{1+\sigma}{2l_nj_{n+1}} 
+\delta_n \varphi(y),
\label{recursion_gen}
%
\end{align}
%
where
%%
\begin{equation}
\label{eq: phi}
\varphi(y):=\sqrt{\frac{\sigma}{2}+y}\left[2 +\log (1+y)\right]^{3/2},
\end{equation}
%
satisfies the bound  \eqref{nonparbound}
for any good block $B$ and $n \geq 1$.
\end{proposition}
%
The proof of this Proposition is postponed to Section \ref{subsec:outline}.
The recursion  \eqref{recursion_gen} between $g_n$ and $g_{n+1}$ is obtained by decomposing
  a path at level $n+1$  
into subbpaths contained in  subblocks of size $K_n$. We then express the total passage time as a maximum of a sum of the partial passage times in each subblock, where the maximum is over all possible intermediate heights of the path at the interfaces. 
The last term $\delta_n \varphi$ of \eqref{recursion_gen} 
is a fluctuation estimate (see Proposition \ref{prop:fluct} below) on the difference between the expectation of the maximum of partial times and the maximum of the expectations. The  remaining part of \eqref{recursion_gen} comes from approximating each partial passage time with its mean and using the induction hypothesis \eqref{nonparbound}. 


\bigskip


{\color{black}
To ge of Proposition \ref{prop_block},
we next bound the functions $g_n$ in terms of the reference function $\tau^{\rho_n,J_n}$ }
%
\begin{equation}
\label{eq: parametric form}
g_n(\sigma, y) \leq \frac{\rho_n^\sigma - \sigma^- +y}{J_n}=\tau^{\rho_n^\sigma,J_n}(\sigma,y)\leq \tau^{\rho_n,J_n}(\sigma,y),
\end{equation}
%
where
%
\be
\label{better_J} 
%
%
J_n :=j_{n+1},
\quad 
\rho_n^\sigma :=\sup_{y\geq \sigma^-} \Big\{
%
j_{n+1} \,g_n(\sigma, y)-y
%
\Big\} + \sigma^-,\quad
%
\rho_n:=\max(\rho_n^1,\rho_n^{-1}).
%
\ee
%
We must then show that $\rho_n$ and $J_n$ satisfy (ii) and (iii) of Proposition \ref{prop_block}.
%
To this end, in Section \ref{sec:other_proofs}, we will prove Propositions \ref{prop:estimation_gen} and \ref{prop: estimation suites} below
(recall that $l_n$, $\Delta_n$ and $\rho_n$ actually depend on $\varepsilon$):
%

\begin{proposition}
\label{prop:estimation_gen}
Assume $\hh$ with $b\in[1,2)$.
%and choose  $\gamma \leq \inf \{ \gamma_0, \frac{2}{b} -1 \}$ (with $\gamma_0$ introduced in \eqref{eq: parameters}), 
Then for $\varepsilon$ small enough,
%there exists $\varepsilon_1>0$ (depending only on $b$, on the constants $c,\beta$ in $\hh$ and on the choice of $J_1$ in \eqref{level_1}), such that for %$\varepsilon\leq\varepsilon_1$,} 
%the sequence $(g_n)_{n \geq 1}$ defined in Proposition \ref{prop:induction_gen}
%satisfies the bound
%
%\begin{equation}
%\label{eq: parametric form}
%g_n(\sigma, y) \leq \frac{\rho_n^\sigma - \sigma^- +y}{J_n}=\tau^{\rho_n^\sigma,J_n}(\sigma,y),
%\end{equation}
%with 
%\be
%\label{better_J} 
%J_n :=j_{n+1}
%\quad \text{and} \quad 
%\rho_n^\sigma :=\sup_{y\geq \sigma^-} \Big\{
%%
%j_{n+1} \,g_n(\sigma, y)-y
%%
%\Big\} + \sigma^-,
%\ee
%
the sequence $(\rho_n^\sigma)_{n\in\N^*}$ defined in \eqref{better_J} satisfies
%
\be
\label{better_recursion}
%
\rho_{n+1}^\sigma \leq \frac{j_{n+2}}{j_{n+1}}\left[
%
\left( 1-\frac{1}{l_n}  \right)
\rho_n^\sigma +\frac{1}{l_n}+\Delta_n
%
\right],
%
\ee
where $\Delta_n=\Delta_n(\varepsilon)$ has the following property: there exist $\varepsilon_1>0$ and $C>0$ such that for every $0<\varepsilon\leq\varepsilon_1$ and $n\geq 1$
%
\be
\label{def_Delta_n}
%
\Delta_n \leq C \, j_{n+1}\frac{\delta_n^2}{2(j_{n+2}^{-1}-j_{n+1}^{-1})}
%
\left[\log\left(
%
\frac{\delta_n}{j_{n+2}^{-1}-j_{n+1}^{-1}}
%
\right)\right]^{3},
%
\ee
with $\delta_n$ as in \eqref{eq: delta n}.
%In the following, we will consider $\rho_n = \max \{ \rho_n^1, \rho_n^{-1} \}$. 
\end{proposition}
%
 Assumption \hh\, ensures that the decay of $j_n$ to $r/4$ is slow enough so that the additional fluctuations of order $\Delta_n$ do not hinder property (ii) of Proposition \ref{prop_block}. 
%Note that 
%(see Proposition \ref{prop: estimation suites} below) 
For the above proposition to be useful, $\gamma$ has to be chosen close to 0 so that the upper bound \eqref{def_Delta_n}
vanishes in the limit $n \to \infty$. 
{\color{black} This will imply Proposition \ref{prop_block}.}
%leads to the following statement:
\begin{proposition}
\label{prop: estimation suites}
Assume $\hh$ with $b\in[1,2)$ and $\gamma <\inf \{ \gamma_0, \frac{2}{b} -1 \}$ where $\gamma_0$ is 
introduced in \eqref{eq: parameters}. 
Then  for small enough $\varepsilon$,  
the sequences $(\rho_n=\rho_n(\varepsilon))_{n\geq 1}$  and $(J_n=J_n(\varepsilon))_{n\geq 1}$ defined in \eqref{better_J}
satisfy the statements of Proposition \ref{prop_block}.
%
%is  such that 
%\be\label{eq:estimation suites_1}
%$\limsup_{n\to\infty}\rho_n^{\textcolor{black}{\sigma}}<1/2$
%\ee
%
%\textcolor{black}{
%%
%and
%%
%\be\label{eq:estimation suites_2}
%%
%$\textcolor{black}{\limsup}_{\varepsilon\to 0}\limsup_{n\to+\infty}\rho_n^{\textcolor{black}{\sigma}}(\varepsilon)\leq\textcolor{black}{\rho_c(0)=}\frac{1}{2}(1-\sqrt{1-r})$.
%%
%\ee
%%
%}
%, \qquad 
%\lim_{n\to\infty} J_n=r/4 \, ,
%with $J_n>r/4$ for all $n\in\N$.
\end{proposition}
%
%\margincom{\textcolor{black}{mis le comm apres la prop pour
%enchainer les 2 prop}}
%Recall that $l_n$, $\Delta_n$ and $\rho_n$ actually depend on $\varepsilon$.
%To conclude the proof of Proposition \ref{prop_block}, 
%we will then show in Section \ref{sec:other_proofs} that
%
\textcolor{black}{
The following result established in Section \ref{sec:other_proofs} will
lead to Proposition \ref{prop_dilute_limit}.}
%
\begin{proposition}
\label{prop:passage_dilute}
{\color{black}
The dilute limit \eqref{dilute_g} holds as the sequence $g_n$ (defined in Proposition \ref{prop:induction_gen}) satisfies
\margincom{\color{red} Il faudrait un $y_1^\sigma$ ?\textcolor{black}{OK}}
\be
\label{dilute_g bis}
\textcolor{black}{\mbox{For }\sigma^-\leq y<\sigma^+y_1^1(0)-\sigma^-y_1^{-1}(0)}, \qquad 
\limsup_{\varepsilon\to 0}\limsup_{n\to+\infty} g_n (\sigma,y) 
%g_{n,\varepsilon}(\sigma,y)
\leq(\sqrt{\sigma+y}+\sqrt{y})^2.
\ee
Recall that $g_n$ depends on $\varepsilon$ through the coarse graining scale.
}
\end{proposition}
%
%%
%%Proposition \ref{prop: estimation suites} will be proved in Section \ref{sec:completion}. Proposition \ref{prop_block} follows from Propositions \ref{prop: induction} and \ref{prop: estimation suites}.\\ \\
%%
\margincom{CH 31/10 supprim\'e la preuve simplifi\'ee}
%The seemingly natural route to derive Proposition \ref{prop:estimation_gen} from Proposition \ref{prop:induction_gen} would be to plug the {\it ansatz}
%$g_n=\tau^{\rho_n,J_n}$ into \eqref{recursion_gen}, and try to obtain an induction on $(\rho_n,J_n)$. In fact, a more subtle treatment is required, because the induction obtained from such an ansatz turns out to be effective only for $b\in(0,1)$, whereas it is not optimal for $b\in[1,2)$.
%%
%Although not necessary, let us indicate how to use the parametric form  when $b\in (0,1)$.
%In this case, Propositions 
%\textcolor{black}{\ref{prop:estimation_gen} and \ref{prop: estimation suites}} can be replaced by the following variant, that is simpler to prove.
 %The price of this simpler approach is to restrict assumption \hh\, to values $b\in (0,1)$.
%%
%%
%\begin{proposition}
%\label{prop: induction}
%%
%Assume $\hh$ with $b \in (0,1)$, and $\gamma <\inf \{ \gamma_0, \frac{1}{b} -1 \}$, with $\gamma_0$ introduced in \eqref{eq: parameters}. Then the following  two assertions hold :\\ \\
%%
%\noindent
%(i) There exists a sequence $\{\rho_n,J_n\}_{n \geq 1}$ with $\rho_1< 1/2$ and $J_1 \in (\textcolor{black}{r/4},1/4)$, which satisfies Proposition \ref{prop_block}, and the recursive bounds 
%\begin{eqnarray}
%%
%J_{n+1} & \geq & \min(J_n,j_{n+1}) - C \delta_n,
%\label{j_one_2}\\
%%
%\rho_{n+1} & \leq & \left( 1-\frac{1}{l_n} \right)
%\rho_n+\frac{1}{l_n} + C \delta_n,
%\label{rho_one_2}
%\end{eqnarray}
%%
%for some constant $C >0$, with $\delta_n$ defined by \eqref{eq: delta n}.\\ \\
%%
%(ii) Any such sequence satisfies $\limsup_{n\to+\infty}\rho_n<1/2$ and $J_n>r/4$.
%%
%%
%%
%\end{proposition}
%%%
%%%
%%%
%
%The derivation of Proposition \ref{prop: induction} is postponed to Subsection \ref{subsec: proof b in (0,1)}. 
%If $b\in[1,2)$, statement (i) of Proposition \ref{prop: induction} can be derived,
%%by using the parametric recursion \eqref{j_one_2}--\eqref{rho_one_2} 
%but the second inequality of
%statement (ii) breaks down and Proposition \ref{prop_block} could not be established this way. In fact, the way \eqref{j_one_2}--\eqref{rho_one_2} is obtained from \eqref{recursion_gen} is not optimal,
%because it amounts to bounding the fluctuation term $\delta_n\sqrt{\sigma/2+y}[2+ \log (1+y)]^{3/2}$ by $\delta_n(1+y)$, which is not accurate for large $y$, that corresponds to densities close to $1/2$ (hence the expected location of the plateau). In \eqref{j_one_2}, fluctuations of $J_n$ are controlled on the one hand by assumption \hh, which guarantees a margin of order $L^{-b/2}$ above $r/4$, and on the other hand by the fluctuations $\delta_n$, which are essentially of order $L^{-1/2}$, where $L$ is the block size. 
%The latter annihilates the former if $b\geq 1$.
%
 





\section{Proof of Proposition \ref{prop:induction_gen}}
\label{subsec:outline}



In this section, we prove the recursion in Proposition \ref{prop:induction_gen}.
To this end, we decompose a path of length $K_{n+1}$ according to its traces on the interfaces between the subblocks of size $K_n$ (see Figure \ref{fig: coarse grained}).
The set of such traces will hereafter be called the ``skeleton'' of the path.
The idea is to use  \eqref{nonparbound} as an induction hypothesis for the subpaths in each block of size $K_n$. If we neglect the fluctuations of these subpaths,
the ```mean'' computation reduces to  optimizing  the positions of the traces so as to maximize the total passage times of subpaths of  level $n$. 
%
%We present here this ``mean'' part of the computation.
%The real proof in the next section involves controlling the fluctuations of the subpaths via concentration inequalities as well as the entropy induced by the many possible skeletons.
This ``mean'' induction relation is altered by an error term (see Proposition \ref{prop:fluct} below) arising from fluctuations of the subpaths as well as the entropy induced by the many possible skeletons. 
%Fluctuations will be controlled by means of concentration inequalities. This eventually leads to a bound of the form \eqref{bound_on_good_blocks} at level $n+1$, %with an induction relation between $\rho_{n+1}$ and $\rho_n$, and $J_{n}=j_{n+1}$. We will show  that the error term does not alter the essential properties (ii), %(iii) in Proposition \ref{prop_block}.\\ \\
%



\subsection{Skeleton decomposition}


\begin{figure}[h]
  \centering
\includegraphics[width=7cm, angle=270]{coarse_graining.pdf}
%\includegraphics[width=8cm, angle=270]{coarse_graining.pdf}
%\def\svgwidth{3cm}
%  \hspace*{.1cm}
%\input{coarse.pdf_t}
\caption{\small A block of level $n+1$ is partitioned into blocks of length $K_n$ (only 3 blocks are depicted). The grey regions represent the boundaries between the blocks of  level $n$ which are separated by a microscopic length 1.
A coarse grained path is depicted and the black dots denote the renewal points $k_i$.} 
  \label{fig: coarse grained} 
\end{figure}




We consider $B=[x,x'=x+K_{n+1}-1]\cap\Z$ a block of order $n+1$, where $x\in\Z$. 
Let $\gamma= ( (x_k,y_k) )_{k=0,\ldots,m-1}$
be a path restricted to $B$ connecting $(x,0)=(x_0,y_0=0)$ to $(x',y'=[K_{n+1}y])=(x_{m-1},y_{m-1})$.
We define the {\it skeleton} $s(\gamma)=\tilde{\gamma}$ of $\gamma$ as follows
(see figure \ref{fig: coarse grained}). 
Let $k_0=-1$ and $y_{-1}=0$. For $i\in\N$, we set
%
\begin{eqnarray*}
%
k_{2i+1} & : = & \min\{k>k_{2i}:\,x_k=x+(i+1)K_n-1\},\\
%
k_{2i+2} & : = & \max\{k\geq k_{2i+1}:\,x_k=x+(i+1)K_n-1\}.
%
\end{eqnarray*}
%
Because $x_{k+1}-x_k\leq 1$, we necessarily have $x_{1+k_{2i+2}}=x+(i+1)K_n$ and $y_{1+k_{2i+2}}=y_{k_{2i+2}}$. Note that $x_{k_{2l_n-1}}=x_{k_{2l_n}}=x'$ and $y_{k_{2l_n}}=y'$.
%
%
Recall that the block $B$ is made of $l_n = \lfloor K_n^\gamma \rfloor$ boxes of length $K_n$.
The skeleton $s(\gamma)$ of $\gamma$ is then the sequence $\tilde{\gamma}=(\tilde{y}_i,\tilde{z}_i)_{i=1,\ldots,l_n}\in(\N^2)^{l_n}$ given by
%
\begin{eqnarray}
%
\tilde{y}_i & := & y_{k_{2i-1}}-y_{k_{2i-2}}\label{skeleton_y},\\
%
\tilde{z}_i & : = & y_{k_{2i}}-y_{k_{2i-1}}\label{skeleton_z}.
%
\end{eqnarray}
%
%
By definition, we have 
%
\be\label{total_height}
\sum_{i=1}^{l_n}\left(\tilde{y}_i+\tilde{z}_i\right)=y'=  \lfloor K_{n+1} y\rfloor.
\ee
%
In a similar way for the paths going from right to left, if $B=[x'=x-K_{n+1}+1,x]\cap\Z$, we may define the skeleton of a path connecting $(x,0)=(x_0,y_0=0)$ to $(x',y'=[K_{n+1}y])=(x_{m-1},y_{m-1})$. 
%
Let $k_0=-1$ and $y_{-1}=-1$. For $i\in\N$, let
%
\begin{eqnarray*}
%
k_{2i+1} & : = & \min\{k>k_{2i}:\,x_k=x-(i+1)K_n+1\} , \\
%
k_{2i+2} & : = & \max\{k\geq k_{2i+1}:\,x_k=x-(i+1)K_n+1\}.
%
\end{eqnarray*}
%
Because $x_{k+1}-x_k\geq -1$, we necessarily have $x_{1+k_{2i+2}}=x-(i+1)K_n$ and $y_{1+k_{2i+2}}=1+y_{k_{2i+2}}$. Note that $x_{k_{2l_n-1}}=x_{k_{2l_n}}=x'$ and $y_{k_{2l_n}}=y'$.
%
%
The skeleton $s(\gamma)$ of $\gamma$ is then the sequence $\tilde{\gamma}=(\tilde{y}_i,\tilde{z}_i)_{i=1,\ldots,l_n}$ given by
\eqref{skeleton_y}--\eqref{skeleton_z}. Since allowed path increments are $(1,0)$ and $(-1,1)$, this sequence must now satisfy the constraint
%$\tilde{y}_1\geq K_n$ and  
$\tilde{y}_i\geq K_n$ for $i\geq 1$. 
\\ \\
%
%
Let $\tilde{\Gamma}_n((x,0),(x',y'))$ denote the set of skeletons of all paths $\gamma$ restricted to $B$ connecting $(x,0)$ and $(x',y')$, that is
the set of sequences $\tilde{\gamma} = (\tilde{y}_i,\tilde{z}_i)_{i=1,\ldots,l_n}\in(\N^2)^{\{1,\ldots,l_n \} }$ satisfying \eqref{total_height}, with the
constraint $\tilde{y}_i\geq K_n$ in the case $x'<x$. We will simply write $\tilde{\Gamma}_n$ when the endpoints are obvious from the context.
%
%
\subsection{passage time decomposition}
%
Let $\sigma = \pm 1$ denote as in \eqref{eq: path direction} the direction of the paths.
%
To encompass both cases $\sigma=\pm 1$, we will use the following simplifying  convention: an interval can be written $[a,b]$ even if $a>b$, in which case it actually means $[b,a]$.
From now on, for notational simplicity, we consider the block $B = [ 0, \sigma(K_{n+1}-1)]$, instead of a block with arbitrary position $x\in\Z$.
For $l\in\{1,\ldots,l_n\}$, we denote by
\textcolor{black}{$B_l:=[\sigma(l-1)K_n,\sigma(lK_n-1)]\cap\Z$} the $l$-th subblock of level $n$ in the decomposition of $B$.
%
For a path skeleton $\tilde{\gamma}=(\tilde{y}_l,\tilde{z}_l)_{l=1,\ldots,l_n}\in\tilde{\Gamma}_n$, define 
%
$$
\tilde{h}_i:=\sum_{j=1}^{i-1}\left[\tilde{y}_j+\tilde{z}_j \right]
% +\frac{1-\sigma}{2}
$$
if $i\geq 2$ and $\tilde{h}_1=0$. The quantity $\tilde{h}_i$ represents the height at which a path with skeleton $\tilde{\gamma}$ enters block $i$ without ever returning to block $i-1$.
%
\begin{figure}[htpb]
%  \centering
%\def\svgwidth{1cm}
%  \hspace*{.1cm}
%\input{independence.pdf_t}
\includegraphics[width=7cm, angle = 270]{independance_blocs.pdf}
%\includegraphics[width=7cm, angle = 270]{independance_blocs.pdf}
\caption{\small A coarse grained path is depicted in a block of order $n+1$. 
The horizontal crossings through each block $B_l$ are restricted to the dark grey regions. The passage time $U^\alpha_B(\sigma,\tilde \gamma)$ depends only on the variables $\{ Y_{i,j} \}$ inside the grey regions which are disjoint from the regions used by the vertical paths contributing to  
$V^\alpha_B(\sigma,\tilde \gamma)$.} 
  \label{fig: independence} 
\end{figure}
%
%On the event $\{s^{-1}(\tilde{\gamma})\cap\Gamma^*\neq\emptyset\}$ (i.e. there is at least one maximizing path with skeleton $\tilde{\gamma}$),
%
For a path $\gamma\in\Gamma_B((0,0),(\sigma(K_{n+1}-1),y'))$  with skeleton $\tilde{\gamma}$, we have that 
%
\be
%
T^\alpha_B(\gamma)\leq U^\alpha_{B}(\sigma, \tilde{\gamma})+V^\alpha_{B}(\sigma, \tilde{\gamma}) \leq T^\alpha_B((0,0),( \sigma (K_{n+1}-1),y' ) \big),
\label{decomp_block}
%
\ee
%
where
%
$$
U^\alpha_{B}(\sigma,\tilde{\gamma})  := 
\sum_{l=1}^{l_n} U^\alpha_{B,l}(\sigma,\tilde{\gamma}),\quad
% =
%\sum_{l=1}^{l_n}
%
%T^\alpha_{B} \big( (\sigma(l-1)K_n,\tilde{h}_l),(\sigma lK_n-1,\tilde{h}_l+\tilde{y}_l) \big)\\
%\label{block_interface} \\
%
V^\alpha_{B}(\sigma,\tilde{\gamma}) :=  
\sum_{l=1}^{l_n} V^\alpha_{B,l}(\sigma,\tilde{\gamma}) ,
% %=
% \sum_{l=1}^{l_n} T^\alpha_B
% \big (  (\sigma lK_n-1,\tilde{h}_l+\tilde{y}_l),
% (\sigma lK_n-1,\tilde{h}_l+\tilde{y}_l+\tilde{z}_l) \big)
%
$$
%
with 
%
\begin{eqnarray}
%
U^\alpha_{B,l}(\sigma,\tilde{\gamma}) & := &
%
T^\alpha_{B_l} \big( \sigma (l-1)K_n,\tilde{h}_l),(\sigma (lK_n-2),\tilde{h}_l+\tilde{y}_l+\sigma-1) \big),
\nonumber \\
%
V^\alpha_{B,l}(\sigma,\tilde{\gamma})  &  := &  T^\alpha_{B} \big( (\sigma(lK_n-1),\tilde{h}_l+\tilde{y}_l+\frac{\sigma-1}{2}),(\sigma (lK_n-1),\tilde{h}_l+\tilde{y}_l+\frac{\sigma-1}{2}+\tilde{z}_l) \big),
\nonumber\\
& & \label{eq: def U V}
%
\end{eqnarray}
%
where $U^\alpha_{B}(\tilde{\gamma})$ is the contribution of the horizontal crossings in the 
blocks $B_l$ and  $V^\alpha_{B}(\tilde{\gamma})$ the contribution of the vertical paths at the 
junction of the blocks $B_l$ (see figure \ref{fig: independence}). 
Noticing that the second inequality in \eqref{decomp_block} is an equality if and only if $\tilde{\gamma}$ is the skeleton
of the optimal path, we get
%
%
\be
%\nonunmber
%
T^\alpha_B((0,0),(\sigma(K_{n+1}-1),y'))  =  \max_{
%
\tilde{\gamma}\in\tilde{\Gamma}_n((0,0),(\sigma(K_{n+1}-1),y'))
%
}\left\{
%
U^\alpha_{B}(\sigma,\tilde{\gamma})+V^\alpha_{B}(\sigma,\tilde{\gamma})
%
\right\}.
%
\label{double_max}
%
\ee
%
To derive Proposition  \ref{prop:induction_gen}, we have to estimate 
\begin{equation}
\label{eq: objectif}
\tau^\alpha_{n+1,B}(\sigma,y)
:= \frac{1}{K_{n+1}} \Exp \Big( T^\alpha_B((0,0),(\sigma(K_{n+1}-1),y')) \Big)
\end{equation}
with  $y' = \lfloor K_{n+1} y\rfloor$ and  we decompose this expectation into the sum of two components:
%
\be\label{mean_opti}
%
\max_{
%
\tilde{\gamma}\in\tilde{\Gamma}_n((0,0),(\sigma(K_{n+1}-1),y'))
%
}
%
\left\{
%
\Exp \left( \frac{U^\alpha_{B}(\sigma,\tilde{\gamma})}{K_{n+1}} \right)
+\Exp \left( \frac{V^\alpha_{B}(\sigma,\tilde{\gamma}) }{K_{n+1}} \right)
%
\right\},
%
\ee
%
that is the ``mean optimization problem" and a  ``fluctuation part'' defined for  
$y' = \lfloor K_{n+1} y\rfloor$ as
%
\begin{eqnarray}
%
& {\mathcal F}_n (y) =  \Exp \left( \frac{1}{K_{n+1}}\max_{
%
\tilde{\gamma}\in\tilde{\Gamma}_n((0,0),(\sigma(K_{n+1}-1),y'))
%
}\Big\{
U^\alpha_{B}(\sigma,\tilde{\gamma})+V^\alpha_{B}(\sigma,\tilde{\gamma})
\Big\}  \right) 
\nonumber\\
%
& \qquad \qquad \qquad \qquad 
-  \max_{ \tilde{\gamma}\in\tilde{\Gamma}_n((0,0),(\sigma(K_{n+1}-1),y'))
%
}
% 
\left\{
%
\Exp \left( \frac{U^\alpha_{B}(\sigma,\tilde{\gamma})}{K_{n+1}} \right)
+\Exp \left( \frac{V^\alpha_{B}(\sigma,\tilde{\gamma}) }{K_{n+1}} \right)
\right\}.
\label{fluct_part}
%
\end{eqnarray}

The term \eqref{mean_opti}, which involves known information from subblocks, will give the main recursion structure, 
while  \eqref{fluct_part} will be an error term. The latter will be controlled by fluctuations and entropy of paths.
%This will generate an additional error term $\delta_n$. 
%in the recursions \eqref{eq: onethird}. 
The precise result that will be established in Section \ref{sec: prop passage time} is the following:%
\begin{proposition}
\label{prop:fluct}
%
With the notation \eqref{fluct_part}, one has uniformly in $y$
%
\begin{eqnarray}
%\nonumber
%%
%&& \Exp\max_{\tilde{\gamma}\in\tilde{\Gamma}_n((0,0),(\sigma(K_{n+1}-1),K_{n+1}y))}K_{n+1}^{-1}\left[
%%
%U^\alpha_{B}(\sigma,\tilde{\gamma})
%+V^\alpha_{B}(\sigma,\tilde{\gamma})
%%
%%
%\right]\\
%& - & \max_{\tilde{\gamma}\in\tilde{\Gamma}_n((0,0),(\sigma(K_{n+1}-1),K_{n+1}y))}\Exp K_{n+1}^{-1}\left[
%%
%U^\alpha_{B}(\sigma,\tilde{\gamma})
%+V^\alpha_{B}(\sigma,\tilde{\gamma})
%%
%%
%\right]
{\mathcal F}_n (y)
\leq
\delta_n\sqrt{\frac{\sigma}{2}+y} \,\big( 1 +\log(1+y) \big)^{3/2},
\label{eq:fluct}
\end{eqnarray}
%
with $\delta_n$ defined in \eqref{eq: delta n}.
%
%\be
%\label{eq: delta n}
%%j_{n+1}:=\frac{r}{4}+\frac{a}{K_{n+1}^{b/2}},\quad\mbox{ and }\quad
%\delta_n:= C\frac{(\ln K_{n+1})^{3/2}}{\sqrt{K_n}}
%\ee
%for some constant $C>0$
%
\end{proposition}

We will in fact replace the upper bound in \eqref{eq:fluct} 
by a slightly worse one
for the sole purpose of making it a concave function of $y$, which is important for us. We therefore observe that
%
\be
\label{concave_bound}
%
\mathcal F_n(y)\leq\mathcal G_n(y):=\delta_n\varphi(y),
%
\ee
%
where $\varphi$ is the (concave) function defined by \eqref{eq: phi}.
%
\subsection{The {\bf main recursion} \eqref{recursion_gen}}
%
Using the skeleton decomposition, we are now going to derive Proposition \ref{prop:induction_gen}.
%
Let us explain the choice \eqref{level_1} of $g_1$ in Proposition \ref{prop:induction_gen}.
To initiate the induction relation, we need a bound at level $1$ for $\rho_1$ and $J_1$. 
For $n=1$, a good block {\color{black} at level 1} contains only rates $\alpha(x) =1$.
Since the restricted passage times are smaller than the unrestricted ones, 
and the latter are superadditive, the asymptotic shape \eqref{def_limheight_k}--\eqref{eq: homogeneous LPP} of the homogeneous last passage percolation 
yields the exact upper bound
$$
%
\tau^\alpha_{1,B}(\sigma,y)\leq\left(\sqrt{\sigma+y}+\sqrt{y}\right)^2 =:g_1(\sigma,y).
$$
%
By  definition, $g_1$ is concave. 
Note that, if $g_n(\sigma,.)$ is   concave, then $g_{n+1}(\sigma,.)$ defined by \eqref{recursion_gen} inherits this property.



\bigskip

Suppose now that the inequality \eqref{nonparbound} 
$$
\tau^\alpha_{n,B}(\sigma,y)\leq g_n(\sigma,y)
$$  
holds at step $n$ and that $g_n$ is concave. We will show that the recursion is valid at step $n+1$ with $g_{n+1}$ defined as in \eqref{recursion_gen}.





We first focus on the mean optimization problem \eqref{mean_opti} and consider a good block $B = [0
, \sigma (K_{n+1} - 1) ]$
at level $n+1$.
For a fixed disorder $\alpha$, by superadditivity and uniformity of $\alpha$ in the $y$-direction,
 recalling \eqref{max_current_lpp}, we have
%
\be
\label{vertical_bound}
%
\Exp\left[V^\alpha_{B,l}(\sigma,\tilde{\gamma})\right]
\leq 
\frac{1}{j_{\infty,B}} \; \tilde{z}_l.
%
\ee
%
Since $B$ is a good block, $j_ {\infty,B}$ satisfies \eqref{condition_good}. 
Thus 
%
\be
\label{eq: def j n+1}
j_{\infty,B}\geq\frac{r}{4}+\frac{a}{K_{n+1}^{b/2}}=:j_{n+1},
\ee
%
(recall that $j_{n+1}$ was introduced in \eqref{condition_good} as one of the conditions defining a good block).
%
%where $j_{n+1}$ was defined in \eqref{eq: delta n}.
%
As $B$ is a good block, the subblocks $B_l$ are good for all values of $l=1,\ldots,l_n$ except for 
possibly one bad subblock with index $i_0$.
The recurrence hypothesis \eqref{nonparbound} at level $n$
implies that the mean passage time on a  good subblock $B_l$ is bounded by 
%
\be
\label{induction}
%
\Exp\left[U^\alpha_{B,l}(\sigma,\tilde{\gamma})\right]= K_n\tau^\alpha_{n,B_l}\left(
%
\sigma,\frac{\tilde{y}_l}{K_n} \right)
\leq 
%
K_n g_n\left(
%
\sigma,\frac{\tilde{y}_l}{K_n} \right).
%
%\frac{1}{J_n} \left[
%%
%%
%K_n\left(
%%
%\rho_n-\frac{1-\sigma}{2}
%%
%\right)+\tilde{y}_l
%%
%%
%\right]
%+c_n(K_n+\tilde{y}_l)\ln\left(
%
%1+\frac{\tilde{y}_l}{K_n}
%
%\right)
%
\ee
%
For the possibly remaining value $i_0$ such that $B_{i_0}$ is a bad block, we use  a crude upper bound 
by artificially extending the path in order to compare its cost to the one of a vertical connection:
%
$$
U^\alpha_{B,i_0}(\sigma,\tilde{\gamma})\leq 
T^\alpha_{B_{i_0}}\left(\left(\sigma(i_0-1)K_n, \tilde{h}_{i_0} \right),\left(\sigma(i_0-1)K_n,
\tilde{h}_{i_0} +\tilde{y}_{i_0}+\frac{1+\sigma}{2}K_n\right)\right),
$$
%
%\margincom{detailler comment on l'obtient}
which yields, as in \eqref{vertical_bound},
%
\be\label{crude_vertical_bound}
%
\Exp\left[U^\alpha_{B,i_0 }(\sigma,\tilde{\gamma})\right]
\leq \frac{1}{j_{\infty,B_{i_0}}} \; \left(\tilde{y}_{i_0}+\frac{1+\sigma}{2}K_n\right)
\leq \frac{1}{j_{n+1}} \left(\tilde{y}_{i_0} +\frac{1+\sigma}{2}K_n\right).
%
\ee
%
%
{\color{black} 
Note that  if there is no bad subblock, we will still apply 
\eqref{crude_vertical_bound} to an arbitrarily chosen subblock to avoid distinguishing this seemingly better case,
which ultimately would not improve our result.}
%
Combining the above expectation bounds, we obtain
%
\be
\label{mean_time_bound}
\Exp\left[
%
U^\alpha_{B}(\sigma,\tilde{\gamma})
+V^\alpha_{B}(\sigma,\tilde{\gamma})
%
%
\right]\leq K_{n+1}  \; g_{n+1}^{(1)}(\sigma,y,\tilde{\gamma}),
%
\ee
%
%
where 
%
\begin{equation}
\label{eq: onethird}
%
g_{n+1}^{(1)}(\sigma,y,\tilde{\gamma}):=
\frac{1}{l_n}\left\{
%
\sum_{l=1,\,l\neq i_0}^{l_n}g_n(\sigma,\bar{y}_l)+\frac{1}{j_{n+1}}\left[
%
\frac{1+\sigma}{2}+\bar{y}_{i_0}+\sum_{l=1}^{l_n} \bar{z}_l
%
\right]
%
%
\right\},
%
%
\end{equation}
%
where $(\bar{y}_l,\bar{z}_l)_{l=1,\ldots,l_n}\in[0,+\infty)^{2l_n}$ is the rescaled skeleton defined by
$\bar{y}_l=K_n^{-1}\tilde{y}_l$ and $\bar{z}_l=K_n^{-1}\tilde{z}_l$, which satisfies the constraint \eqref{total_height}, whence
%
\be\label{continuum_skeleton}
%
\sum_{l=1}^{l_n}\left(
%
\bar{y}_l+\bar{z}_l
%
\right)
\leq l_ny
\quad \text{with} \quad 
\bar{y}_l\geq\sigma^- .
%
\ee
%
Define
\begin{equation}
\label{eq: bar y}
\sigma^-\leq\, \bar{y}:=\frac{1}{l_n-1} \sum_{l=1,\ldots,l_n:\,l\neq i_0}\bar{y}_l
\leq  \frac{l_n}{l_n-1} y,
\end{equation}
%
so that from \eqref{continuum_skeleton}, we have
%
\be
\label{total_height_macro}
\bar{y}_{i_0}+\sum_{l=1}^{l_n}\bar{z}_l
%=K_n^{-1}[K_{n+1} y]
\leq l_n y-(l_n-1)\bar{y} .
\ee
%
%
By concavity of $g_n$, \eqref{mean_time_bound}--\eqref{eq: onethird} and \eqref{total_height_macro}, we obtain an upper bound for \eqref{mean_opti}:
%
\begin{align}
%
\label{mean_time_bound_gen}
%
& \max_{
%
\tilde{\gamma}\in\tilde{\Gamma}_n((0,0),(\sigma(K_{n+1}-1),[K_{n+1}y]))
%
}
%
\left\{
%
\Exp \left( \frac{U^\alpha_{B}(\sigma,\tilde{\gamma})}{K_{n+1}} \right)
+\Exp \left( \frac{V^\alpha_{B}(\sigma,\tilde{\gamma}) }{K_{n+1}} \right)
%
\right\}\\
& \qquad \qquad\qquad \leq  \sup_{\sigma^- \leq \bar{y}\leq \frac{l_n}{l_n -1} y}
\left\{ 
\left(1-\frac{1}{l_n}\right)
\left[
g_n(\sigma,\bar{y}) - \frac{\bar{y}}{j_{n+1}} 
%
\right] \right\}
+ \frac{y}{j_{n+1}} + \frac{1+\sigma}{2l_nj_{n+1}} ,
\nonumber
\end{align}
where the value of $\bar y$ in \eqref{eq: bar y} has been replaced by a supremum.
To bound from above 
$\tau^\alpha_{n+1,B}(\sigma,y)$ (see  \eqref{eq: objectif}), it is enough to combine 
\eqref{mean_time_bound_gen} and Proposition \ref{prop:fluct}.
This completes the proof of Proposition \ref{prop:induction_gen}.









%
%\begin{proposition}\label{prop:induction_gen}
%%
%Let 
%%
%%\be
%%\label{eq: delta n}
%$$
%j_{n+1}:=\frac{r}{4}+\frac{a}{K_{n+1}^{b/2}}
%%,\quad\mbox{ and }\quad
%%\delta_n:= C\frac{(\ln K_{n+1})^{3/2}}{\sqrt{K_n}}
%%\ee
%$$
%%for some constant $C>0$. 
%Then
%%
%the sequence $(g_n)$ defined on $[\sigma^-,+\infty)$ by 
%%
%\be\label{level_1}
%g_1=\tau^{\rho_1,J_1},\quad\mbox{with } J_1\in(r/4,1/4),\,\rho_1=2J_1
%\ee
%%
%and
%\margincom{{\bf CH} longueur du chemin vers $(x,y)$ est $x+2y$, donc variance $\sim\sqrt{x/2+y}$}
%%
%\begin{eqnarray}
%%
%%
%g_{n+1}(\sigma,y)  & :=  & \sup_{0\leq\bar{y}\leq \frac{l_n}{(l_n-1)} y}
%\left\{
%\left(1-\frac{1}{l_n}\right)\left[
%%
%g_n(\sigma,\bar{y})- \frac{\bar{y}}{j_{n+1}}
%%
%\right] \right\} 
%+ \frac{y}{j_{n+1}}+  \frac{1+\sigma}{2l_nj_{n+1}}\nonumber\\
%%
%%
%& + & \delta_n\sqrt{\frac{\sigma}{2}+y}[1+\ln(1+y)]^{3/2},
%\label{recursion_gen}
%%
%\end{eqnarray}
%%
%satisfies the bound \eqref{nonparbound}.
%%
%%with $j_n$ and $\delta_n$ as in \eqref{eq: delta n},
%%
%%
%\end{proposition}
%%
%%
%Note that, if $g_n(\sigma,.)$ is assumed concave, then $g_{n+1}(\sigma,.)$ defined by \eqref{recursion_gen} inherits this property.
%Let us explain the choice \eqref{level_1} of $g_1$ in Proposition \ref{prop:induction_gen}.
%\textcolor{black}{To initiate} the induction relation, we need a bound at level $1$ for $\rho_1$ and $J_1$. \margincom{\bf CH}For $n=1$, a good block of order 1 contains only rates $\alpha(x) =1$.
%Since the restricted passage times are smaller than the unrestricted ones, the
%values for $\rho_1$ and $J_1$ can be inferred from the asymptotic shape of the homogeneous last passage percolation \eqref{eq: homogeneous LPP}
%and the corresponding exact upper bound for the expected passage time, due to superadditivity:
%%
%$$
%%
%\tau^\alpha_{1,B}(\sigma,y)\leq\left(\sqrt{\sigma+y}+\sqrt{y}\right)^2
%\leq 
%4\left(\frac{1}{2}\sigma+y\right)
%\leq \tau^{2J_1,J_1}(\sigma,y)
%%
%$$
%%
%for every $\sigma\in\{-1,1\}$ and any $J_1\in(0,1/4]$. From \eqref{ref_time}, inequality \eqref{bound_on_good_blocks} holds with any pair
%%
%\be\label{level_1}
%%
%{\color{black} 
%\frac{r}{4} <J_1 < \frac{1}{4}, \quad \rho_1=2J_1 
%}
%%
%\ee
%%
%With this choice, $\rho_1 < 1/2$. Given this initialization, we will show in Section \ref{sec:completion} that
%%
%%
%\begin{proposition}\label{prop:estimation_gen}
%%
%%Assume $b\in[1,2)$. Then
%The sequence $(g_n)$ defined in Proposition \ref{prop:induction_gen} satisfies the bound
%%
%$g_n\leq\tau^{\rho_n,J_n}$ with 
%%
%\be\label{better_J} J_n=j_{n+1}\ee
%%
%and $\rho_n$ defined inductively by
%%for some sequences $(\rho_n)$ and $(J_n)$ satisfying the properties of Proposition \ref{prop_block}.
%%
%\be\label{better_recursion}
%%
%\rho_{n+1}=\frac{j_{n+2}}{j_{n+1}}\left[
%%
%\left(
%%
%1-\frac{1}{l_n}
%%
%\right)\rho_n+\frac{1}{l_n}+\Delta_n
%%
%\right]
%%
%\ee
%%
%where $\Delta_n=\Delta_n(\varepsilon)$ has the following property: there exist $\varepsilon>0$ and $C'>0$ such that, for every $0<\varepsilon\leq\varepsilon_0$
%and $n\geq 1$,
%%%
%\be\label{def_Delta_n}
%%
%\Delta_n\leq C' j_{n+1}\frac{\delta_n^2}{2(j_{n+2}^{-1}-j_{n+2}^{-1})}\left[
%%
%\log\left(
%%
%\frac{\delta_n}{j_{n+2}^{-1}-j_{n+1}^{-1}}
%%
%\right)
%%
%\right]
%%
%\ee
%%
%\end{proposition}
%%
%The role of Assumption \hh\mbox{ }is to ensure that the decay of $j_n$ to $r/4$ is slow enough that the additional fluctuations of order $\Delta_n$ do not hinder property (ii) of Proposition \ref{prop_block}. 
%%
%%
%Recall that $l_n$, $\Delta_n$ and $\rho_n$ actually depend on $\varepsilon$.
%To conclude the proof of Proposition \ref{prop_block}, we will show that
%%%
%\begin{proposition}
%\label{prop: estimation suites}
%%Assume $b\in(0,1)$. Then 
%Assume $b\in[1,2)$. Then under Assumption (H), for small enough $\varepsilon$, 
%a sequence $\big(\rho_n)_{n\geq 1}$ satisfying 
%%the inequalities
%%\eqref{j_one_2} and \eqref{rho_one_2} 
%%
%\eqref{better_recursion} will be such that 
%$$
%\limsup_{n\to\infty}\rho_n<1/2
%$$
%%, \qquad 
%%\lim_{n\to\infty} J_n=r/4 \, ,
%%with $J_n>r/4$ for all $n\in\N$.
%\end{proposition}
%%%
%%%Proposition \ref{prop: estimation suites} will be proved in Section \ref{sec:completion}. Proposition \ref{prop_block} follows from Propositions \ref{prop: induction} and \ref{prop: estimation suites}.\\ \\
%%%
%%
%\textcolor{black}{
%%%%
%Note that a separate study of the case $b\in(0,1)$ is not necessary: indeed, if assumption (H) is satisfied for some $b\in(0,1)$, it is satisfied {\it a fortiori} for $b=1$. However, we note that when $b\in(0,1)$, Proposition \ref{prop:estimation_gen} could be replaced by the following variant, that is simpler to prove.
%%Though not strictly necessary, we shall provide this simplified version before the general one, because
%%we think it gives simpler insight into our approach.
%%
%%These propositions will be proved respectively in Sections \ref{sec: prop passage time} and \ref{sec:completion}.\\ \\
%%
%In this case, we simply plug the ansatz $g_n(\sigma,y)=\tau^{\rho_n,J_n}(\sigma,y)$ into the recursion \eqref{recursion_gen} so that the latter yields a well-behaved  recursion on $(\rho_n,J_n)$:
%%
%%
%\begin{proposition}
%\label{prop: induction}
%%
%\mbox{}\\ \\
%%
%(i) There exists a sequence $\{\rho_n,J_n\}_{n \geq 1}$ with $\rho_1< 1/2$ and $J_1 \in (0,1/4)$, which satisfies Proposition \ref{prop_block}, and the recursive bounds 
%\begin{eqnarray}
%%
%J_{n+1} & \geq & \min(J_n,j_{n+1})-B\delta_n
%\label{j_one_2}\\
%%
%\rho_{n+1} & \leq & \left(
%%
%1-\frac{1}{l_n}
%%
%\right)\rho_n+\frac{1}{l_n}+B\delta_n
%\label{rho_one_2}
%%
%\end{eqnarray}
%%
%for some constant $B>0$, with $\delta_n$ defined by \eqref{eq: delta n}.\\ \\
%%
%(ii) Any such sequence satisfies $\limsup_{n\to+\infty}\rho_n<1/2$ and $J_n>r/4$.
%%
%%such that the statements of Proposition \ref{prop_block} hold.
%%
%%
%\end{proposition}
%%%
%%%
%%%
%%Then we can show  as in Prop \ref{prop: estimation suites} that the recursions \eqref{j_one_2}--\eqref{rho_one_2}, together with the initial values \eqref{level_1}, imply properties (ii) and (iii) in Proposition \ref{prop_block}. 
%If $b\in[1,2)$, this procedure still yields Proposition \ref{prop: induction}, but
%statement (ii) of Proposition \ref{prop_block} fails. In fact, the way \eqref{j_one_2}--\eqref{rho_one_2} is obtained from \eqref{recursion_gen} is not optimal,
%because it amounts to bounding the fluctuation term $\delta_n\sqrt{\sigma/2+y}[\ln(1+y)]^{3/2}$ by $\delta_n(1+y)$, which is not accurate for large $y$, that corresponds to densities close to $1/2$ (hence the expected location of the plateau). In \eqref{j_one_2}, fluctuations of $J_n$ are controlled on the one hand by assumption (H), which guarantees a margin of order $L^{-b/2}$ above $r/4$, and the fluctuations $\delta_n$, which are essentially of order $L^{-1/2}$, where $L$ is the block size. 
%The latter annihilates the former if $b\geq 1$.
%%
%}
%
\section{Consequences of the main recursion}
\label{sec:other_proofs}
%
In this section, we prove Propositions \ref{prop:estimation_gen}, \ref{prop: estimation suites} 
and  \ref{prop:passage_dilute}.


\subsection{Proof of Proposition \ref{prop:estimation_gen}}\label{subsec:conseq_1}
%\margincom{{\bf Updated by Thierry}}
%
As $g_1(\sigma,.)$ is concave, 
the recursion \eqref{recursion_gen} implies that $g_{n}(\sigma,.)$ is a concave function for all $n$. 
For notational simplicity, we shall write details of the proof for $\sigma=1$.  In this case, we simply write $g_n(.)$ for $g_n(\sigma,.)$ and $\rho_n$ for $\rho_n^\sigma$.
We will only briefly indicate what changes are involved for $\sigma=-1$.
We consider the sequence $(g_n)_{n \geq 1}$ given by the  recursion \eqref{recursion_gen} and set
%
\be
\label{def_yn}
y_n:=\inf\left\{
%
y\geq 0:\quad 
g'_n(y) \leq  \frac{1}{j_{n+1}} 
%
\right\},
\ee
%
where $g'_n$ stands for the right derivative of the concave function. Thus, if $y\geq(1-l_n^{-1})y_n$
%
\begin{equation}
%
 g_{n+1}(y) =  \left(
%
1-\frac{1}{l_n}
%
\right)\left[
%
g_n\left(y_n\right)- \frac{y_n}{j_{n+1}}
%
\right]
%
 +  \frac{y}{j_{n+1}} + \frac{1}{l_nj_{n+1}} +\delta_n \varphi(y),
\label{gn_1} 
%
\end{equation}
%
and if $y\leq(1-l_n^{-1})y_n$
%
\begin{equation}
 g_{n+1}(y) =  \left(
%
1-\frac{1}{l_n}
%
\right)\left[
%
g_n \left( \frac{l_n}{l_n-1} y \right)-\frac{l_n}{l_n-1} \frac{y}{j_{n+1}}
%
\right] +
%
 \frac{y}{j_{n+1}} + \frac{1}{l_nj_{n+1}} +\delta_n \varphi(y).
%
\label{gn_2}
\end{equation}
%
%
\begin{lemma}
\label{lemma_yn}
%
Assume $\hh$ with $b\in[1,2)$. Then for $\varepsilon$ small enough, the sequence $(y_n)_{n\geq 1}$ satisfies
%
\be
\label{value_yn}
\forall n \geq 2, \qquad 
 y_{n-1} \leq y_n  < \infty
\quad \text{and} \quad
%
{\varphi'(y_n)}=\frac{
%
j_{n+1}^{-1}-j_{n}^{-1}
%
}{\delta_{n-1}},
%
\ee
%
with $\varphi$ as in \eqref{eq: phi}.
%
\end{lemma}
%
\begin{proof}
The proof of  \eqref{value_yn} is split in 3 steps.

\smallskip

\noindent
{\it Preliminary computations.} For $n\geq 1$, we set
\begin{equation}
\label{eq: tn}
t_{n+1}:=
%
\frac{j_{n+2}^{-1}-j_{n+1}^{-1}}{\delta_{n}}=\psi_3(K_n),
%
\end{equation}
%
with
%
\begin{eqnarray*}
%
\psi_3(K) & := & (1+\gamma)^{-3/2} \frac{K^{1/2}}{
(\log K)^{3/2}}\left\{
%
\left(
%
\frac{r}{4}+\frac{a}{K^{\frac{b}{2}(1+\gamma)^2}}
%
\right)^{-1}-\left(
%
\frac{r}{4}+\frac{a}{K^{\frac{b}{2}(1+\gamma)}}
%
\right)^{-1}
%
\right\}\\
%
& \stackrel{K\to+\infty}{\sim} & (1+\gamma)^{-3/2}\frac{16a}{r^2
}K^{\frac{1}{2}-\frac{b}{2}(1+\gamma)}
\stackrel{K\to+\infty}{\longrightarrow} 0.
%
\end{eqnarray*}
Since $b\geq 1$ 
{\color{black} and $K_1(\varepsilon)$ diverges in the dilute limit  \eqref{conclusion_lemma_good}}, we conclude that 
%
\be\label{conclude_tn}
\lim_{\varepsilon\to 0}\sup_{n\geq 1}t_{n+1}(\varepsilon)=0.
\ee
%

\noindent
{\it 
%Proof of \eqref{value_yn} for
Case $n=2$.}
%
Since $y_1>(1-l_1^{-1})y_1$, $g'_2(1,y_1)$ is obtained by differentiating \eqref{gn_1}:
$$
g'_2(1,y_1)-j_{3}^{-1}={j_2}^{-1}-j_3^{-1}+\delta_1\varphi'(y_1)=\psi_4(K_1),
$$
%
where $\psi_4(K_1)>0$, for large $K_1$, because as $K_1\to+\infty$, we have
%
$$
j_2^{-1}-j_3^{-1}\sim -K_1^{-b(1+\gamma)/2}
\quad \text{with} \quad 
\frac{b(1+\gamma)}{2} > \frac{1}{2}
\quad \text{and} \quad 
\delta_1=C\frac{(\log K_{1})^{3/2}}{K_1^{1/2}},
$$
%
{\color{black}
(recall  $b\geq 1$ and $\gamma>0$). Thus $y_2>y_1>(1-l_1^{-1})y_1$ as $g'_2(1,y_1)>j_{3}^{-1}$ for $\varepsilon$ small enough.}
%
Hence, for $y$ in the neighborhood of $y_2$, $g'_2(1,y)$ is also obtained by differentiating the expression \eqref{gn_1}.
It follows that
%
$$
y_2=\inf\left\{
%
y\geq 0:\qquad \varphi'(y)\leq \frac{j_3^{-1}-j_2^{-1}}{\delta_1}
%
\right\}.
$$
%
Since $\varphi$ is strictly concave and $\lim_{y\to +\infty}\varphi'(y)=0$,  \eqref{conclude_tn} implies that for $\varepsilon$ small enough, $y_2$ is the unique solution of $\varphi'(y_2)=t_2$.
%
%
Thus identity \eqref{value_yn} holds for $n=2$.

\medskip

\noindent 
{\it Case $n>2$.} We are going to prove the  claim  by induction.
Suppose that \eqref{value_yn} is valid up to rank $n$. 
To show $y_{n+1} \geq y_n$, it is enough to check that
$$
g'_{n+1}(y_n)>j_{n+2}^{-1}.
$$
%
Since $y_n>(1-l_n^{-1})y_n$, the above derivative is computed from the expression \eqref{gn_1}. Thus, using the induction hypothesis \eqref{value_yn}, we get for $n\geq 2$
%
\begin{eqnarray}\nonumber
g'_{n+1}(y_n)-j_{n+2}^{-1} & = &  j_{n+1}^{-1}-j_{n+2}^{-1}+{\delta_n}\varphi'(y_n)
%\\
%%
%%
%\nonumber & = & 
= j_{n+1}^{-1}-j_{n+2}^{-1}+\frac{\delta_n}{\delta_{n-1}}\left(j_{n+1}^{-1}-j_n^{-1}\right)\\
%
& = & \psi(K_{n-1}) ,
\label{induction_gn}
%
\end{eqnarray}
%
 where, since $K_{n}=K_{n-1}^{1+\gamma}$,
%
\begin{eqnarray}
%
\nonumber\psi(K) & = & \left(
%
\frac{r}{4}+\frac{a}{K^{\frac{b}{2}(1+\gamma)^2}}
%
\right)^{-1}-\left(
%
\frac{r}{4}+\frac{a}{K^{\frac{b}{2}(1+\gamma)^3}}
%
\right)^{-1}\\
%
& + & (1+\gamma)^{3/2}K^{-\gamma/2}\left[
%
\left(
%
\frac{r}{4}+\frac{a}{K^{\frac{b}{2}(1+\gamma)^2}}
%
\right)^{-1}-\left(
%
\frac{r}{4}+\frac{a}{K^{\frac{b}{2}(1+\gamma)}}
%
\right)^{-1}
%
\right].
\label{def_psi}
%\\ 
%%
%& = & \psi_1(K)+\psi_2(K)\label{def_psi}
%
\end{eqnarray}
%
Let us respectively denote by $\psi_1(K)$ and $\psi_2(K)$ the first and second line on the r.h.s. of \eqref{def_psi}.
Then as $K\to+\infty$, 
%
$$\psi_1(K)\sim-16ar^{-2}K^{-b(1+\gamma)^2/2},\quad
\psi_2(K)\sim16a(1+\gamma)^{3/2} \, r^{-2}K^{-b(1+\gamma)/2-\gamma/2}.
$$
%
Since for $b\geq 1$ and $\gamma>0$ we have
%
$$
\frac{b}{2}(1+\gamma)+\frac{\gamma}{2}<\frac{b}{2}(1+\gamma)^2.
$$
%
It follows that $\psi(K)>0$ for $K$ large enough. 
%Since $\lim_{\varepsilon\to 0}K_1(\varepsilon)=\infty$
As $K_1(\varepsilon)$ diverges when  $\varepsilon$ tends to 0 (see  \eqref{conclusion_lemma_good}), 
we have that for small enough $\varepsilon$,
$y_{n+1} \geq y_n\geq(1-l_n^{-1})y_n$ holds for all $n\geq 2$.\\ \\
%
%
As $g'_{n+1}(y_{n+1})$ is given by the derivative of \eqref{gn_1} and $\varphi$ is strictly concave, we have to solve
\begin{equation}
\label{eq: solve yn+1}
g'_{n+1}(y_{n+1})=j_{n+1}^{-1}+{\delta_n}\varphi'(y_{n+1})=j_{n+2}^{-1}
\quad \Rightarrow \quad
\varphi'(y_{n+1})= \frac{j_{n+2}^{-1} - j_{n+1}^{-1}}{\delta_n} = t_{n+1}.
\end{equation}
%
%To do this, we set
%\begin{equation}
%\label{eq: tn}
%t_{n+1}:=
%%
%\frac{j_{n+2}^{-1}-j_{n+1}^{-1}}{\delta_{n}}=\psi_3(K_n),
%%
%\end{equation}
%%
%with
%%
%\begin{eqnarray*}
%%
%\psi_3(K) & := & \textcolor{black}{(1+\gamma)^{-3/2}}\frac{K^{1/2}}{%(1+\gamma)^{3/2}
%(\log K)^{3/2}}\left\{
%%
%\left(
%%
%\frac{r}{4}+\frac{a}{K^{\frac{b}{2}(1+\gamma)^2}}
%%
%\right)^{-1}-\left(
%%
%\frac{r}{4}+\frac{a}{K^{\frac{b}{2}(1+\gamma)}}
%%
%\right)^{-1}
%%
%\right\}\\
%%
%& \stackrel{K\to+\infty}{\sim} & \textcolor{black}{(1+\gamma)^{-3/2}}\frac{16a}{r^2%(1+\gamma)^{3/2}
%}K^{\frac{1}{2}-\frac{b}{2}(1+\gamma)}
%\stackrel{K\to+\infty}{\longrightarrow} 0.
%%
%\end{eqnarray*}
%Since $b\geq 1$, we conclude that $\lim_{n\to+\infty}t_n=0$ and  a solution $y_{n+1} < \infty$ of 
%\eqref{eq: solve yn+1} always exists.
%
As above, \eqref{conclude_tn} implies that, for $\varepsilon$ small enough, a solution of \eqref{eq: solve yn+1} exists for all $n\geq 2$. This proves the second part of the claim \eqref{value_yn}.  The proof is similar for $\sigma=-1$.
%
%
%In the case $\sigma=-1$, the infimum \eqref{def_yn} defining $y_n$ is over $y\geq 1$. The initialization step differs slightly since we then have
%%
%$$
%y_1=\frac{(1-\rho_1)^2}{1-2\rho_1}
%$$
%%
%and $y_1=1$. Therefore the inequality in \eqref{value_yn} is no longer trivial for $n=2$. However, since
%$\varphi'(y_2)=t_2$, this inequality will be true if $t_2\leq\varphi'(1)$, which holds for $\varepsilon$ small enough thanks to \eqref{conclude_tn}.
%
%
\end{proof}
%
%\begin{corollary}\label{corgen}
%%
%For $b\in[1,2)$, the sequence $(\rho_n)_{n\geq 1}$ defined by
%%
%$$
%\rho_n:=\sup_{y\geq 0}[
%%
%j_{n+1}g_n(y)-y
%%
%]
%$$
%%
%%then $\limsup_{n\to+\infty}\rho_n<1/2$. 
%%
%satisfies the recursion \eqref{better_recursion}.
%%
%\end{corollary}
%%
%The above corollary completes the proof of 
%%\eqref{prop_block} 
%Proposition \ref{prop:estimation_gen} in the case $b\in[1,2)$, since we have by definition
%%
%$$
%g_n(y) \leq \frac{\rho_n+y}{j_{n+1}}=\tau^{\rho_n,j_{n+1}}(1,y)
%$$
%
%where $j_{n+1}>r/4$ and $\lim_{n\to+\infty}j_{n+1}=r/4$.
%
Using Lemma \ref{lemma_yn}, 
%as announced in Subsection \ref{subsec:coarse}, 
we can now  complete the proof of Proposition 
\ref{prop:estimation_gen}.
We must show that  inequality \eqref{better_recursion} holds for the sequence $(\rho_n)_{n\in\N^*}$ with $\Delta_n$ satisfying \eqref{def_Delta_n}.
%Since $\lim_{n\to+\infty}j_n=r/4$, we have 
%%
%\be\label{limsn}
%%
%\limsup_{n\to+\infty}\rho_n=\frac{r}{4}\limsup_{n\to+\infty}s_n,
%%
%\ee
%%
%where
%%
%$$
%s_n:=j_{n+1}^{-1}\rho_n=\sup_{y\geq 0}[
%%
%g_n(y)-j_{n+1}^{-1}y
%%
%].
%$$
%
%
By definition \eqref{def_yn} of $y_n$, 
{\color{black} the supremum in \eqref{better_J} is reached at $y_n$ so that} 
%
$$
\rho_n=j_{n+1}g_n(y_n)-y_n.
$$
%
We are going to obtain a recursion for $\rho_n$. To this end, consider
%
$$
%
\rho_{n+1}=j_{n+2}g_{n+1}(y_{n+1})-y_{n+1}.
%
$$
%
By Lemma \ref{lemma_yn}, $y_{n+1} \geq y_n>(1-l_n^{-1})y_n$, so $g_{n+1}(y_{n+1})$ is obtained from \eqref{gn_1}. Thus
%
\begin{eqnarray}
%
\rho_{n+1} & =  j_{n+2}\left(1-\frac{1}{l_n}\right)\left[
%
g_n(y_n)- \frac{y_n}{j_{n+1}}
%
\right]+ \frac{j_{n+2}}{l_n \, j_{n+1}}+ \left( \frac{j_{n+2}}{j_{n+1}} - 1 \right)  y_{n+1}
+ j_{n+2}\delta_n\varphi(y_{n+1})\nonumber\\
%
& \leq  \frac{j_{n+2}}{j_{n+1}} \left( \left(1-\frac{1}{l_n}\right)\left[
%
j_{n+1}g_n(y_n)- {y_n}
%
\right]+ \frac{1}{l_n} +j_{n+1}\delta_n\varphi(y_{n+1})
\right),
\label{sn_part_one}
%
\end{eqnarray}
%
where on the second line we have used $j_{n+2}\leq j_{n+1}$. Setting ${\Delta}_n:=j_{n+1}\delta_n\varphi(y_{n+1})$, we recovered the inequality  \eqref{better_recursion}, and it remains to verify \eqref{def_Delta_n}.
%
Starting from 
$$
\varphi'(y)\stackrel{y\to+\infty}{\sim}
\frac{1}{2\sqrt{y}}
(\log y)^{3/2},
$$
%
we see that
\be
\label{phi_prime}
%
\varphi[\varphi^{'-1}(t)]\stackrel{t\to 0}{\sim}\frac{1}{2t}
%
\left(\log\frac{1}{4t^2}\right)^3.
\ee
%
Recall that 
%
by \eqref{eq: solve yn+1}, $y_{n+1}=\varphi^{'-1}(t_{n+1})$, where
$t_n$ is defined by \eqref{eq: tn} and satisfies \eqref{conclude_tn}.
%%$\lim_{n\to+\infty}t_n=0$. Since $\lim_{\varepsilon\to 0}K_1(\varepsilon)=+\infty$ and $K_{n+1}(\varepsilon)\geq K_n(\varepsilon)$ for all $n\geq 1$, 
Thus,
%
there exist $C',C''>0$ and
$\varepsilon_2>0$ such that, for every $0<\varepsilon\leq\varepsilon_2$ and $n \geq 1 $
%
$$
\varphi(y_{n+1})=\varphi[\varphi^{'-1}(t_{n+1})]\leq C''\frac{1}{t_{n+1}} \, \big|\log t_{n+1} \big|^{3}
%
\leq C' \frac{\delta_n}{2(j_{n+2}^{-1}-j_{n+1}^{-1})}
%
\left[
%
\log\left(
%
\frac{\delta_n}{j_{n+2}^{-1}-j_{n+1}^{-1}}
%
\right)
%
\right]^3.
%
$$
%
This implies \eqref{def_Delta_n} with  ${\Delta}_n=j_{n+1}\delta_n\varphi(y_{n+1})$.


\medskip

For $\sigma = -1$, 
still writing $\rho_n$ for $\rho_n^{\sigma}$, we have
$$\rho_n - 1= \sup_{y \geq 1}
\big\{ j_{n+1} g_n(-1,y) - y \big\}$$
and we get a  recursion similar to \eqref{sn_part_one}   
\begin{eqnarray*}
%
\rho_{n+1} - 1 \leq  \frac{j_{n+2}}{j_{n+1}} \left( \left(1-\frac{1}{l_n}\right)\left[
%
\rho_n - 1
%
\right]  + j_{n+1}\delta_n\varphi(y_{n+1})
\right),
\end{eqnarray*}
which can be rewritten
\begin{eqnarray*}
%
\rho_{n+1} \leq  \frac{j_{n+2}}{j_{n+1}} \left( \left(1-\frac{1}{l_n}\right) \rho_n 
+ \frac{1}{l_n} +
j_{n+1}\delta_n\varphi(y_{n+1})
+ \frac{j_{n+1}}{j_{n+2}} -1
\right).
\end{eqnarray*}
For $b \geq 1$, the remainder $\frac{j_{n+1}}{j_{n+2}} -1$ can be bounded by $\Delta_n$ so that the same type of inequality is also valid for $\sigma = -1$. 




\subsection{Proof of Proposition \ref{prop: estimation suites}}
\label{proof_seq}
%
%\begin{proof}
Let $a_n=1-\frac{1}{l_n}$. Then one can see by induction that 
 \eqref{better_recursion} implies
%
\begin{eqnarray}
%
\rho_n^\sigma & \leq & \rho_1^\sigma \prod_{i=1}^{n-1}a_i+\left(
%
1-\prod_{i=1}^{n-1}a_i
%
\right)
+\sum_{i=1}^{n-1}\Delta_i\prod_{j=i+1}^{n-1}a_j\nonumber\\
%
& \leq & \rho_1^\sigma \prod_{i=1}^{n-1}a_i+\left(
%
1-\prod_{i=1}^{n-1}a_i
%
\right)
+\sum_{i=1}^{n-1}\Delta_i ,
\label{induction_rho}
\end{eqnarray}
where we used that $\frac{j_{i+1}}{j_i}\leq 1$ for any $i \geq 1$.
%
%
Remember that the quantities $\rho_n$, $j_n$, $a_n$, $\Delta_n$ actually depend on $\varepsilon$.
%
%
Since $g_1$ is given by \eqref{level_1}, a simple computation shows that
%
\begin{equation}\label{eq:rho1}
\rho_1^1 :=\sup_{y\geq 0} \Big\{ j_2 \, g_1(1,y)-y \Big\} 
\end{equation}
%
is the smaller root $\rho$ of the equation
%
\begin{equation}\label{eq:j2}
\rho(1-\rho)=j_2,
\end{equation}
%
and that the supremum in (\ref{eq:rho1}) is achieved at 
{\color{black} 
$y_1^1:=\frac{\rho_1^2}{1-2\rho_1}.$
For $\sigma = -1$, we have
}
\begin{equation}
\label{eq:rhoprime1}
\rho_1^{-1}:=\sup_{y\geq 1} \Big\{ j_2 \, g_1(-1,y)-y \Big\}+1=\rho_1^1
\end{equation}
%
and the supremum achieved for {\color{black} $y_1^{ -1}:=\frac{(1-\rho_1)^2}{1-2\rho_1}
$.
}
%
In particular, \textcolor{black}{since} {\color{black} the divergence of   $K_1$ \eqref{conclusion_lemma_good}}
  implies $\lim_{\varepsilon\to 0}j_2(\varepsilon)=r/4$, we also have
%
\begin{equation}
\label{rho1_dilute}
%
\lim_{\varepsilon\to 0}\rho_1^\sigma (\varepsilon)=\frac{1}{2}\left(1-\sqrt{1-r}\right)=\rho_c(0)
%
\end{equation}
%
that is the lower solution of (\ref{eq:j2}) with $r/4$ instead of $j_2$.
This says that  the approximation   after one step of  renormalization is close to the dilute limit.
{\color{black} Lemma \ref{lemma_induction}, stated below, shows that 
 $\rho_n(\varepsilon)$ remains close to $\rho_c(0)$ for $\varepsilon$ small.}
By (\ref{rho1_dilute}), \eqref{induction_rho} and Lemma \ref{lemma_induction} below, we have
%
\begin{equation}
\label{wehave}
\limsup_{\varepsilon\to 0}\limsup_{n\to+\infty}\rho_n(\varepsilon)
\leq\rho_c(0).
\end{equation}
{\color{black} This completes the proof of Proposition \ref{prop: estimation suites}.}
%
%
\begin{lemma}
\label{lemma_induction} 
Assume $\hh$ with $b\in[1,2)$ and
{\color{black}  $K_1$ satisfies \eqref{conclusion_lemma_good}.}
%
With the notation of Lemma \ref{lemma_good}, we fix 
$\gamma < \min \big\{ \gamma_0, \frac{2}{b} -1 \big\}$.
Then 
%for $b \in [1,2)$
%
\begin{enumerate}
\item[(1)] $\lim_{\varepsilon\to 0}\prod_{n=1}^{+\infty} a_n(\varepsilon)=1$,
\item[(2)] $\lim_{\varepsilon\to 0}\sum_{n=1}^{+\infty}\Delta_n(\varepsilon)=0$.
\end{enumerate}
%
\end{lemma}
%
%
%Since 
%%
%$$
%\prod_{n=1}^{+\infty} a_n(\varepsilon)\leq\exp\left[
%%
%-\sum_{n=1}^{+\infty}\frac{1}{l_n(\varepsilon)}
%%
%\right]
%$$
%%
\begin{proof}
\ \\
\noindent
{\it Proof of (1).} We have to show that
%
\be\label{limit_1}
\lim_{\varepsilon\to 0}\sum_{n=1}^{+\infty} \log \left(1-\frac{1}{l_n(\varepsilon)}\right)=0 .
\ee
%
%
Since 
%
$$
l_n(\varepsilon)=\exp\left[
%
\log \big(K_1(\varepsilon) \big) \, \gamma(1+\gamma)^{n-1}
%
\right]\geq \exp\left[
%
\gamma(1+\gamma)^{n-1}
%
\right],
$$
%
we have, for $n\geq 2$,  $l_n(\varepsilon)^{-1}\leq C(\gamma):=e^{-\gamma(1+\gamma)}<1$. Hence, for $n\geq 2$,
%
$$
0\leq -
\log \left(1-\frac{1}{l_n(\varepsilon)}\right)\leq \frac{1}{l_n(\varepsilon)}+\frac{C'(\gamma)}{l_n(\varepsilon)^2}
%
\leq (1+C'(\gamma))\exp\left[
%
-\gamma(1+\gamma)^{n-1}
%
\right].
%
$$
%
The limit \eqref{limit_1} then follows from dominated convergence, and $\lim_{\varepsilon\to 0}K_1(\varepsilon)=+\infty$, which implies $\lim_{ \varepsilon\to 0}l_n(\varepsilon)=0$ for any $n\geq 1$.\\ \\
%
{\it Proof of (2).} Here we  can write  $\Delta_n \simeq \psi_0(K_n)$, where
%
\begin{eqnarray}
& {\psi}_0(K)  := \frac{(\log K)^3}{K}\left[
%
\left(\frac{r}{4}+
%
\frac{a}{K^{\frac{b}{2} (1+\gamma)^2}}
%
\right)^{-1}
%
- \left(\frac{r}{4}+
%
\frac{a}{K^{\frac{b}{2}  (1+\gamma)}}
%
\right)^{-1}
%
\right]^{-1} \nonumber \\
%
& \qquad \qquad \qquad \qquad  \times  
%
\left(
%
\log\left\{
%
\frac{(\log K)^{3/2}}{\sqrt{K}}
%
\left[
%
\left(\frac{r}{4}+
%
\frac{a}{K^{\frac{b}{2} (1+\gamma)^2}}
%
\right)^{-1}
%
- \left(\frac{r}{4}+
%
\frac{a}{K^{\frac{b}{2}  (1+\gamma)}}
%
\right)^{-1}
%
\right]^{-1}
%
\right\}
%
\right)^3
%
\label{eq: b<2}\\
%
& \qquad \qquad 
\stackrel{K\to+\infty}{\sim} %  \left[\frac{b}{2}(1+\gamma)-\frac{1}{2}\right]
C'' (\log K)^{6} K^{\frac{b}{2}(1+\gamma)-1}.
\nonumber
\end{eqnarray}
%
%
%We see that for $b<2$ we can choose $\gamma$ small enough so that 
%
for some constant $C''>0$. The assumption on $b$ and the choice of $\gamma$ imply that
$c:=1-\frac{b}{2}(1+\gamma)>0$
(Equation \eqref{eq: b<2} is the main reason for restricting to the case $b<2$). 
By \eqref{conclusion_lemma_good}, there exists $\varepsilon_1>0$ such that $K_n(\varepsilon)\geq 2$ for  every $n\geq 1$ and $\varepsilon\in[0,\varepsilon_1]$.
Thus, by \eqref{eq: b<2}, there exists a constant $D>0$ such that, for such $n$ and $\varepsilon$,
%
$$
\Delta_n(\varepsilon)\leq \frac{D}{K_n(\varepsilon)^{c}}\leq\frac{D}{K_1(\varepsilon)^{c(1+\gamma)^n}}\leq \frac{D}{2^{c(1+\gamma)^n}} .
$$
%
%
Since $\lim_{\varepsilon\to 0}K_1(\varepsilon)=+\infty$, the result follows again from dominated convergence.
\end{proof}




\subsection{Proof of Proposition \ref{prop:passage_dilute}}



Note that 
\begin{equation}
\label{dilute_product}
%
\lim_{\varepsilon\to 0}
\prod_{n=1}^{+\infty}\frac{l_n(\varepsilon)}{l_n(\varepsilon)-1}=1.
%
\end{equation}
{\color{black} 
Thus for any $\delta>0$, there exists $\varepsilon_*>0$ such that, for $\varepsilon \leq \varepsilon_*$, the following holds
%
\begin{equation}
\label{eq:epstar}
\prod_{n=1}^{+\infty}\frac{l_n(\varepsilon)}{l_n(\varepsilon)-1}<1+\delta ,
\end{equation}
and $(y_n)_{n\geq 0}$ is an increasing sequence thanks to Lemma \ref{lemma_yn}.
We fix $y < \frac{y_1}{1+\delta}$ and $\varepsilon \leq \varepsilon_*$.
For any $N \in \mathbb{N}^*$, we define the sequence 
\begin{equation}
\label{finiteprod_dilute_k}
y_{N,N}:=y 
\quad \text{and} \quad 
\forall n \in \{ 1, N-1 \}, \qquad y_{n,N}:=
\prod_{k= n + 1}^N \frac{l_k}{l_k-1} y  \leq y_1 \leq y_n.
\end{equation}
As $\frac{l_n}{l_n-1}y_{n,N}=y_{n-1,N}\leq y_n$, then 
 $g_{n+1}(1,y_{n,N})$ is determined by \eqref{gn_2} so that 
\begin{eqnarray}
\label{recursion}
%
g_{n+1}(1,y_{n,N}) = \left(
%
1-\frac{1}{l_n}
%
\right)g_n\left(
%
1,
%\frac{l_n}{l_n-1} 
y_{{n-1,N}}
%
\right)+\frac{1}{l_n}{j_{n+1}}+\delta_n\varphi\left(y_{{{n,N}}}\right).
%
\end{eqnarray}
Starting from $y_{N,N}:=y $ and proceeding recursively, we deduce that 
\begin{align} 
g_{N+1}(1,y) \leq  \prod_{n=1}^N\left(1-\frac{1}{l_n}\right)g_k\left[
%
1,\prod_{n=1}^N\frac{l_n}{l_n-1}y
%
\right]
%\nonumber\\
% & + & 
+\frac{4}{r}\sum_{n=1}^N 
%\prod_{r =n+1}^N\frac{l_r}{l_r-1} }
\frac{1}{l_n}
+\sum_{n=1}^N\delta_n\varphi \left(   
\prod_{r =n+1}^N\frac{l_r}{l_r-1}y 
\right) .
\label{eventually}
\end{align}
From \eqref{rho1_dilute}, we know that $y_1 = y_1 (\varepsilon)$ converges to $y_1(0)
=\frac{\rho_c(0)^2}{1-2\rho_c(0)}$. Furthermore  $\lim_{\varepsilon\to 0}l_n(\varepsilon)=+\infty$ and $\lim_{\varepsilon\to 0}\delta_n(\varepsilon)=0$. Thus  it follows from (\ref{eventually}) that 
%
\begin{equation}
\label{lim_gN}
\forall y< \frac{y_1(0)}{1+\delta}, \qquad 
\limsup_{\varepsilon\to 0} \limsup_{N\to+\infty}g_N(1,y){\leq}g_1(y) .
\end{equation}
In the dilute limit,  $\delta$ can be  arbitrarily small so that the inequality above holds more generally for $y<y_1 (0)$.
A similar result holds for $\sigma = -1$.}





\section{Fluctuation bounds : Proof of Proposition \ref{prop:fluct}} 
\label{sec: prop passage time}


{\color{black}
Proposition \ref{prop:fluct} is proved in this section. 
Preliminary estimates are stated in Subsection \ref{subsec:conc} and then applied in Subsection \ref{fluct_ent}, which is the body of the proof.
}


\subsection{Concentration estimates}
\label{subsec:conc}
%
%
We shall need  a classical gaussian concentration inequality for last passage times. In the following lemma, it is assumed that the service times $Y_{i,j}$  involved in the definition \eqref{def_passage}--\eqref{def_last} of last passage times
are i.i.d. random variables  {\it bounded} by $M$ instead of being exponentially distributed. To avoid confusion with the previous notation, the corresponding probability $\Prob_M$ and expectation $\Exp_M$ are denoted below by an index $M$.
%
\begin{lemma}\cite[Lemma 3.1]{mar}
\label{lemma_concentration}
%
Assume that $Y=(Y_{i,j}:\,(i,j)\in\Z\times\N)$ is a vector of non negative independent random variables bounded from above by $rM$.
%
% ajouté r ici le 16/2
%
Let $(x_1,y_1)$ and $(x_2,y_2)$ in $\Z\times\N$ be such that $(x_2-x_1,y_2-y_1) \in \mathcal W$. Then 
%
\begin{eqnarray}\nonumber
T^\alpha \big( (x_1,y_1),(x_2,y_2) \big)  =  \Exp_M\left[
%
T^\alpha \big( (x_1,y_1),(x_2,y_2) \big)
%
\right]
%
+  8M\sqrt{L((x_1,y_1),(x_2,y_2))}Z,
\label{concentration_inequality}
%
%\ee
\end{eqnarray}
%
where $L((x_1,y_1),(x_2,y_2)):=(x_2-x_1)+2(y_2-y_1)$ is the length of any path connecting $(x_1,y_1)$ to $(x_2,y_2)$, and $Z$ is a random variable with subgaussian tail
$$
\forall t \geq 0, \qquad \Prob_M(|Z|\geq t)\leq \exp(-t^2) \, .
$$
%
\end{lemma}
%
We stress the fact that Gaussian bounds on last passage times are by no means optimal in the case of exponential service times, for which more refined (but also more specific)  gaussian-exponential estimates are available (see e.g. \cite{tal}). However, for our purpose, they have the advantage of being both simple and sufficient,
while also extending to service distributions with heavier tails, as a result of the cutoff procedure introduced in Subsection \ref{fluct_ent}. 

\medskip

The above concentration inequality will be combined with the following result,  established in Appendix \ref{app:proof_cor_max}.
\begin{lemma}\label{cor_max}
%
Let $\mathcal A$ and $\mathcal I$ be finite sets. Assume that for each $a\in\mathcal A$, we have a 
%finite set $\mathcal I_a$, and a 
family 
%
$({\mathcal Y}_{a,i})_{i\in \mathcal I}$ of independent random variables such that, for every $i\in\mathcal I$,
%
\be
\label{hyp_normal}
{\mathcal Y}_{a,i}  \, = \,
%\Exp Y_\alpha
\Exp \big( {\mathcal Y}_{a,i} \big) + \sqrt{V_{a,i}} Z_{a,i},
\ee
%
where $V_{a,i}>0$, and $Z_{a,i}$ is a random variable such that 
%
\be
\label{normal_tail}
\Prob(Z_{a,i}\geq t)\leq e^{-t^2},
\ee
%
for every $t\geq 0$. Then
%
\begin{eqnarray}
%
\Exp \left( \max_{a\in\mathcal A}\sum_{i\in\mathcal I} {\mathcal Y}_{a,i} \right) 
& \leq & 
\max_{a\in\mathcal A}\sum_{i\in\mathcal I} 
\Exp \big( {\mathcal Y}_{a,i}\big)  \nonumber\\
%\Exp Y_\alpha\leq 
%&+ & 
& &  +
%\sigma
\left(
%
\max_{a\in\mathcal A}\sum_{i\in\mathcal I}V_{a,i}
%
\right)^{\frac{1}{2}}
\left(
%
\sqrt{\pi}\sqrt{|\mathcal I|}+\sqrt{\pi}\sqrt{A}+\sqrt{A}\sqrt{\log|\mathcal A|}
%
\right),
\label{exp_max}
%
\end{eqnarray}
%
where $|.|$ denotes  the cardinality, and $A$ is the same constant as in (ii) of Lemma \ref{lemma_normal}.
\end{lemma}







\subsection{Path renormalization: fluctuation and entropy}
\label{fluct_ent}
%
We now proceed in three steps. 
%
In step one, we define a cutoff procedure for the service times $Y_{i,j}$, by conditioning on their maximum,  in order to replace them with bounded variables, to which the  results of Subsection \ref{subsec:conc} apply.
%
In step two, we apply Lemma \ref{cor_max} to passage times in subblocks. This yields for the cutoff service times a result similar to the statement of proposition \ref{prop:fluct},
but without the whole logarithmic correction.
%
%
Finally, in step three, we remove the cutoff and use a bound on the expectation of the maximum of exponential variables, to obtain a quasi-gaussian estimate with a logarithmic correction.\\ \\ 
%
{\bf Step 1.} {\it Notation and conditional measure.}
%
Pick $\gamma$ such that 
%
\be\label{choose_gamma}
0<\gamma<\min \big\{ \gamma_0,(2/b)-1 \big\} ,
\ee
%
with $\gamma_0$ introduced in Lemma \ref{lemma_good}, and $b$ in  \eqref{condition_h_2} and
\eqref{condition_h}.
Let $B=\Z\cap[0,\sigma(K_{n+1}-1)]$ be a block of order $n+1$
and partition $B$ into  subblocks of  level $n$ denoted by
$B_l=[\sigma (l-1)K_n,\sigma (lK_n-1)]\cap\Z$, where $l=1,\ldots,l_n$.

\medskip

Set $y'= \lfloor K_{n+1}y\rfloor$, 
$\tilde{\Gamma}_n=\tilde{\Gamma}_n((0,0),(\sigma(K_{n+1}-1),y'))$
and define 
%$M_n(y')\in\N\backslash\{0\}$ denote a cutoff value for individual times $Y_{i,j}$, and 
%
$$
%E_n(y'):=\{
%
M_B(y):=
%
\max \big \{ Y_{i,j}:\,i\in B,\,j=0,\ldots,y'= \lfloor K_{n+1}y\rfloor  \big  \} .
$$
%
Given $M>0$, denote by $\Prob_{B,M,y'}$ the distribution of $(X_{i,j}:\,\,i\in B,j=0,\ldots,y')$, where $X_{i,j}$ are i.i.d. and have the same distribution as  $Y_{i,j}$ conditioned on $Y_{i,j}\leq rM$.
(Note that after conditioning by $rM$, the percolation paths have weights $\frac{Y_{i,j}}{\alpha} \leq M$ as $\alpha \geq r$).
Denote by $\Prob'_{B,M,y'}$ the distribution of $(Y_{i,j}:\,i\in B,\,j=0,\ldots,y')$ conditioned on $M_B(y)=rM$. 
A vector $(Y'_{i,j}:\,(i,j)\in B\times\{0,\ldots,y'\})$ with distribution $\Prob'_{B,M,y'}$ is obtained as follows. Pick a uniformly distributed $(i_0,j_0)$ in $B\times\{0,\ldots,y'\}$; then give value $rM$ to $Y'_{i_0,j_0}$, and let the other $Y'_{i,j}$ for $(i,j)\neq (i_0,j_0)$ be independent with the same distribution as the above $X_{i,j}$.
%A coupling argument shows that the distribution of $T^\alpha_B((0,0),(\sigma(K_{n+1}-1),y'))$ under $\Prob'_{B,M,y'}$ is stochastically dominated by the distribution of $T^\alpha_B((0,0),(\sigma(K_{n+1}-1),y'))+M$ under $\Prob_{B,M,y'}$.\\ \\
%
%
%
%On the event $E_n(y')$, $T^\alpha_B((0,0),(K_{n+1}-1,y'))$ coincides with the cutoff last passage time
%$T^\alpha_{B,M_B(y)}((0,0),(K_{n+1}-1,y'))$ defined by replacing r.v.'s $Y_{i,j}$ with $Y_{i,j}\wedge M_B(y)$ in 
%$B\times(\Z\cap[0,y'])$, and similarly, every passage time restricted to any of the sub-blocks coincides with its cutoff approximation.
%

\bigskip

\noindent
{\bf Step 2.} {\it Fluctuation and entropy bounds.}  
%
Given $\alpha$, the random variables $\{ U^\alpha_{B,l'}(\sigma,\tilde{\gamma}),$ $V^\alpha_{B,l}(\sigma,\tilde{\gamma}) \}_{l,l'}$ (defined in \eqref{eq: def U V}) are independent under $\Prob_{B,M,y'}$, because they depend on disjoint subvectors of $Y$ (see figure \ref{fig: independence}). 
On the other hand, by Lemma \ref{lemma_concentration}, we get
\begin{eqnarray}
\label{concentration_h}
\begin{cases}
U^\alpha_{B,l}(\sigma,\tilde{\gamma}) 
=
\Exp_M\left[
%
U^\alpha_{B,l}(\sigma,\tilde{\gamma})
%
\right]+8M\sqrt{\sigma K_n+2\tilde{y}_l} \, Z_l^{(1)},
\\
V^\alpha_{B,l}(\sigma,\tilde{\gamma}) 
=
\Exp_M\left[ V^\alpha_{B,l}(\sigma,\tilde{\gamma})\right]+8M\sqrt{2\tilde{z}_l} \, Z_l^{(2)},
\end{cases}
%
%
\end{eqnarray}
%
where $(Z_l^{(i)})_{l=1,\ldots,l_n;i=1,2}$ is a family of r.v.'s independent under $\Prob_{B,M,y'}$ and such that
%
\begin{equation}
\label{eq: one sided}
\Prob_{B,M,y'}\left( Z_l^{(i)}\geq t \right)\leq \exp( -t^2) ,
\end{equation}
%
for all $t\geq 0$. To apply Lemma \ref{cor_max} to the random variables in \eqref{concentration_h},
 we take 
$\mathcal A=\tilde{\Gamma}_n \big((0,0),(\sigma(K_{n+1}-1),y') \big)$ with $\mathcal I=\{1,\ldots,2l_n\}$,
and for $a=\tilde{\gamma}\in \mathcal A$, we set 
$$
l\in\{1,\ldots,l_n\},  \qquad {\mathcal Y}_{a,2l-1}=U^\alpha_{B,l}(\sigma,\tilde{\gamma})
\quad \text{and} \quad 
{\mathcal Y}_{a,2l}=V^\alpha_{B,l}(\sigma,\tilde{\gamma}).
$$ 
Thus in \eqref{exp_max} we have $|\mathcal I|=2l_n=2K_{n+1}/K_n$, and (cf. \eqref{concentration_h} and \eqref{total_height})
%
$$
\sum_{i\in\mathcal I}V_{a,i}=64M^2 (\sigma K_{n+1}+2y')\leq 64M^2 K_{n+1}(\sigma+2y).
$$
%
To estimate the cardinality $|\mathcal A|$ of the skeletons, we need the following 
%
\begin{lemma}
\label{count_skeleton}
For every $y'\in\N$, one has
%
$$
\log | \mathcal A| = 
\log \big |  \tilde{\Gamma}_n  \big( (0,0),(\pm(K_{n+1}-1),y')  \big)  \big |
\leq
2\frac{K_{n+1}}{K_n} \, \left[ 1+\log \left( 1+{K_n y} \right) \right].
$$
%
%
\end{lemma}
%
\begin{proof}
The number of such skeletons satisfies the inequality
%
\be
\label{skeleton_number}
%
\sigma\in\{-1,1\}, \qquad 
\Big| \tilde{\Gamma}_n \big( (0,0),(\sigma(K_{n+1}-1),y')  \big) \Big|
\leq {2l_n+y'-1 \choose 2l_n-1}.
\ee
%
The previous upper  bound follows by noticing that choosing a skeleton amounts to choosing
$2l_n-1$ heights corresponding to the different renewal times to reach the total height $y'$. In fact, 
when $\sigma=1$, some of these heights can be equal if $\tilde y_i =0$ or $\tilde z_i =0$ for some $i 
\leq 2l_n-1$. Thus, the number of ways \textcolor{black}{for} choosing the heights is bounded by the number of ways \textcolor{black}{for} choosing $2l_n-1$
items from a set of $2l_n+y'-1$ items.
%To take this into account,  it is enough to choose a collection of distinct heights in the larger set $2l_n+y'-1$.
Estimate \eqref{skeleton_number} is actually an equality if $\sigma=1$.


%
Recall the inequality:
\be
\label{binomial_bound}
\log \left( \ba{l}
N\\
k
\ea
%
%
\right)\leq N h\left(
%
\frac{k}{N}
%
\right),
\ee
%
where $h$ is defined on $[0,1]$ by
%
$$
-h(x):=x \log x+(1-x) \log (1-x)
{\color{black}
\quad \text{with} \quad h(0) = h(1) = 0.}
$$
Furthermore
$$
u h(1/u) \leq 1+ \log u
$$
%
for $u\geq 1$, and 
%
$$
\frac{2l_n+y'-1}{2l_n-1}\leq 1+\frac{y'}{l_n}
= 1 + \frac{K_{n+1}}{l_n} y,
$$
%
(the inequality follows from $l_n\geq 1$). This completes the  proof of Lemma \ref{count_skeleton}.

\medskip

Bound \eqref{binomial_bound} follows e.g. from Cramer's exact large deviation upper bound applied to a sum of i.i.d. Bernoulli variables with parameter $1/2$ denoted by $(\zeta_i)_{i=1,\ldots,n}$, since
for $k\geq N/2$,
%
$$
\left(
%
\ba{l}
%
N\\
%
k
%
\ea
%
%
\right)
\leq 2^N\Prob\left( \frac{1}{N} \sum_{i=1}^N \zeta_i \geq \frac{k}{N}
\right).
$$
%
%
\end{proof}
%
Combining \eqref{exp_max} with the entropy estimate of Lemma \ref{count_skeleton}, we obtain 
%
\begin{align}
\label{eq:fluct:3}
&\Exp_M \left( \max_{\tilde{\gamma}\in\tilde{\Gamma}_n((0,0),(\sigma(K_{n+1}-1),[K_{n+1}y]))}
\Big\{
%
U^\alpha_{B}(\sigma,\tilde{\gamma})
+V^\alpha_{B}(\sigma,\tilde{\gamma})
%
%
\Big\} \right)\\
& \quad \leq  \max_{\tilde{\gamma}\in\tilde{\Gamma}_n((0,0),(\sigma(K_{n+1}-1),[K_{n+1}y]))}
\Big\{
\Exp_M\left(
%
U^\alpha_{B}(\sigma,\tilde{\gamma})
+V^\alpha_{B}(\sigma,\tilde{\gamma})
%
%
\right)
\Big\}
\nonumber\\
%\end{eqnarray}
%
%is bounded above by 
%
%\be\label{upperbound_cutoff} M\sqrt{A}\sqrt{ K_{n+1}( \sigma +2y)}
%
& \quad + 8M\sqrt{K_{n+1}}\sqrt{\sigma+2y}
\sqrt{ 2\frac{K_{n+1}}{K_n} }
\left(
%
\sqrt{\pi}
%
+\sqrt{A}+
%
\sqrt{A}
%
\sqrt{ \left[ 1+\ln\left( 1+{K_n y} \right) \right] }
%
%%
\right)\nonumber,
%%
%
%
%\ee
%
\end{align}
where we used that $2\frac{K_{n+1}}{K_n}  \geq \pi$.


\medskip
%
\noindent
{\bf Step 3.} {\it  Removing the cut-off on $Y$.}
%
The random variables  $\{ U^\alpha_{B,l}(\sigma,\tilde{\gamma}), \; V^\alpha_{B,l}(\sigma,\tilde{\gamma}) \}_{l,l'}$ are nondecreasing functions of $Y=(Y_{i,j}:\,(i,j)\in\Z\times\N)$ with respect to the product order.
Therefore, their distributions under $\Prob_{B,M,y'}$ are stochastically dominated by their distributions under $\Prob$ and one has 
%if we replace $Y$ by its cutoff $Y\wedge M_B(y)$, we obtain cutoff versions 
%$U^\alpha_{B,l,n}(\tilde{\gamma})$ and $V^\alpha_{B,l,n}(\tilde{\gamma})$, still independent, and stochastically dominated by 
%the original r.v.'s. Correponding sums will be denoted by $U^\alpha_{B,n}(\tilde{\gamma})$ and $V^\alpha_{B,n}(\tilde{\gamma})$.  
%
\be
\label{eq: comparaison E et EM}
\Exp \left[U^\alpha_{B,l}(\sigma,\tilde{\gamma}) \right]
\geq 
\Exp_M \left[U^\alpha_{B,l}(\sigma,\tilde{\gamma}) \right],
\qquad 
\Exp \left[V^\alpha_{B,l}(\sigma,\tilde{\gamma})\right]
\geq 
\Exp_M \left[V^\alpha_{B,l}(\sigma,\tilde{\gamma})\right]  .
\ee
%
On the other hand, a coupling argument shows that the distribution of 
$T^\alpha_B \big( (0,0),(\sigma(K_{n+1}-1),y') \big)$ under $\Prob'_{B,M,y'}$ is stochastically dominated by the distribution of $T^\alpha_B \big( (0,0),(\sigma(K_{n+1}-1),y') \big) + M$ under $\Prob_{B,M,y'}$. 
This property combined with \eqref{eq: comparaison E et EM} and \eqref{eq:fluct:3} yields 
\begin{align}
\label{eq:fluct:3bis}
& \Exp'_M \left( \max_{\tilde{\gamma}\in\tilde{\Gamma}_n((0,0),(\sigma(K_{n+1}-1),[K_{n+1}y]))}
\Big\{ U^\alpha_{B}(\sigma,\tilde{\gamma}) +V^\alpha_{B}(\sigma,\tilde{\gamma}) \Big\}
\right)\\
%
& \qquad \leq  \max_{\tilde{\gamma}\in\tilde{\Gamma}_n((0,0),(\sigma(K_{n+1}-1),[K_{n+1}y]))}
\Big\{
\Exp\left[
%
U^\alpha_{B}(\sigma,\tilde{\gamma})
+V^\alpha_{B}(\sigma,\tilde{\gamma})
\right]  \Big\}+M\nonumber\\
%\end{eqnarray}
%
%is bounded above by 
%
%\be\label{upperbound_cutoff} M\sqrt{A}\sqrt{ K_{n+1}( \sigma +2y)}
%
& \qquad +  8M K_{n+1}\sqrt{\sigma+2y} \sqrt{ \frac{2}{K_n} } 
\left(
\sqrt{\pi}
+\sqrt{A}+
\sqrt{A}
\sqrt{
%%
 \left[
%
%
1+\log \left(
%
1+{K_n y}
%
\right)
%
\right]
%
%
}
\right).\nonumber
\end{align}
%
%The bounds on the passage time were obtained under the conditional measure in order to cut-off the large fluctuations of the environment $Y$.
Recall that $\Exp'_M$ on the left-hand site of \eqref{eq:fluct:3bis} stands for the expectation with respect to $\Prob$ conditioned on the maximum $M_B(y)=M$.
We can now remove this conditioning by integrating both sides of \eqref{eq:fluct:3bis} with respect to the law of $M_B(y)$. We first write
%
\be\label{def_expmax}
%
\Exp \big[ M_B(y) \big]=m\left([yK_{n+1}]K_{n+1}\right),
%
\ee
%
where the function $t\in[0,+\infty)\mapsto m(t)$ is defined as the expectation of the maximum of $1+[t]$ i.i.d. exponential variables of rate 1. In particular, we have
%
\be\label{bound_expmax}
%
m(t)\leq C[1+\log(1+t)],
%
\ee
%
for some constant $C>0$.
%
Thus, after conditioning on $M_B(y)$, we obtain
%
%
%
%\begin{eqnarray}\label{after_conditioning}
%%
%\Exp\left[
%%
%\frac{1}{K_{n+1}}T^\alpha_B((0,0),(\sigma(K_{n+1}-1),y'))
%%
%\right]  & \leq &  g_{n+1}^{(1)} (\sigma,y)+
%%
%m\left(yK_{n+1}\right)\Delta_n(y)
%%\\
%%
%%& =: & \tau^\alpha_{n+1,B,+}(y)
%%
%\end{eqnarray}%
%
\begin{align}
\label{after_conditioning}
& \frac{1}{K_{n+1}}
\Exp \left( \max_{\tilde{\gamma}\in\tilde{\Gamma}_n((0,0),(\sigma(K_{n+1}-1), [ K_{n+1}y ]))}
\Big\{
%
U^\alpha_{B}(\sigma,\tilde{\gamma})
+V^\alpha_{B}(\sigma,\tilde{\gamma})
\Big\} \right) \\
& \quad \leq  \frac{1}{K_{n+1}}
\max_{\tilde{\gamma}\in\tilde{\Gamma}_n((0,0),(\sigma(K_{n+1}-1),[K_{n+1}y]))} 
\Big\{ 
\Exp\left( 
%
U^\alpha_{B}(\sigma,\tilde{\gamma})
+V^\alpha_{B}(\sigma,\tilde{\gamma})
%
%
\right) \Big\}
\nonumber\\
%\end{eqnarray}
%
%is bounded above by 
%
%\be\label{upperbound_cutoff} M\sqrt{A}\sqrt{ K_{n+1}( \sigma +2y)}
%
%& + & 
& \qquad \qquad 
+m\left([yK_{n+1}]K_{n+1}\right)\Delta_n(y) , \nonumber
\end{align}
%
where 
%
\begin{eqnarray*}
%
\tilde \Delta_n(y) := \frac{1}{K_{n+1}}
+
8 \frac{ \sqrt{\sigma+2y}}{\sqrt{K_{n}}} 
\left(
\sqrt{A}\sqrt{\pi}
+ \sqrt{2\pi}
+  \sqrt{A} \sqrt{ 1+\log \left( 1+{K_n y} \right) }
\right) .
%
\end{eqnarray*}
%
A simple computation shows that 
%
$$
m \big([yK_{n+1}]K_{n+1}\big) \; \tilde \Delta_n(y)
\leq
\delta_n\sqrt{\sigma/2+y} \, \big[1+\log (1+y) \big]^{3/2} ,
$$
%
%the second term on the r.h.s. of \eqref{after_conditioning} can indeed be bounded by $\delta_n\sqrt{\sigma/2+y}[\ln(1+y)]^{3/2}$, that is the the r.h.s. of %\eqref{eq:fluct}.
with $\delta_n$ given by \eqref{eq: delta n}.
Using the notation of \eqref{fluct_part}, we get 
$$
\mathcal F_n (y) \leq
\delta_n\sqrt{\frac{\sigma}{2} +y} \;  [1+\log (1+y)]^{3/2} .
%\leq \delta_n\sqrt{ \frac{\sigma}{2} +y}  \; [2 + \log (1+y)]^{3/2} .
$$ 
This completes the proof of Proposition \ref{prop:fluct}.
\qed



\section{ Completion of proofs of Theorems \ref{th_plateau} and \ref{th_dilute_limit}}
% of Theorem \ref{th_plateau}}
\label{sec:completion}


In this section, we complete the remaining parts in the proof of Theorem \ref{th_plateau}.
In Subsection \ref{unrestricted}, we deduce from Proposition \ref{prop_block} a similar statement for unrestricted 
passage times (that is, when the paths are not restricted to the box defined by the endpoints). Finally, 
Theorem \ref{th_plateau} is completed, in Subsection \ref{conclude}, using the fact that most boxes are good.
The dilute limit (Theorem \ref{th_dilute_limit}) is studied in Section \ref{sec:proof_dilute}.



\subsection{Bounds on unrestricted passage times}
\label{unrestricted}

To obtain Theorem \ref{th_plateau} from Proposition \ref{prop_block}, we first
deduce from Proposition \ref{prop_block} the following result for unrestricted passage times, i.e. passage times obtained by maximizing over paths not bound to stay
in the interval between the two endpoints (see figure \ref{fig: unrestricted}).
\begin{figure}[htpb]
\centering
\includegraphics[width=.7\columnwidth]{unrestricted.pdf}
%\includegraphics[width=.7\columnwidth]{unrestricted.pdf}
\caption{\small The optimal path is not restricted to the box $[0,K_n] \times [0,K_n y]$ but can wander around the whole parallelogram marked by the dotted line. The optimal path is split into 3 parts 
$(0,y_{k_1})$, $(y_{k_1},y_{k_2})$ and $(y_{k_2}, (K_n,  \lfloor K_n y\rfloor))$.
 } 
\label{fig: unrestricted} 
\end{figure}


\margincom{\textcolor{black}{c'est un peu genant d'avoir $\sigma$ en indice par intermittence ?}\textcolor{black}{c'etait $\rho_n$ en fait, j'ai corrige la suite}}
Given the sequence {$(\rho_n)_{n \geq 1}$ of Proposition \ref{prop: estimation suites}, we set 
%
\begin{equation}
\label{eq: rhoc}
\overline{\rho}_c=\overline{\rho}_c(\varepsilon) : = \limsup_{n \to \infty} \rho_n\in[0,1/2).
\end{equation}
{\color{black} In the rest of this section, the asymptotics $\lim_{n\to+\infty}$ means that 
we restrict to a subsequence (fixed once and for all) of $(\rho_n)_{n\geq 1}$ that achieves the $\limsup$ in \eqref{eq: rhoc}.}
%
\begin{corollary}
\label{cor_block}
%
%
For $\sigma = \pm 1$ and $y\geq \sigma^-$, we consider the unrestricted passage time
%
\be\label{def_unrestricted}
{\tau}_n^\alpha(\sigma,y):=\Exp \left[ \frac{1}{K_n}T^\alpha(\sigma K_n,[K_ny]) \right].
\ee
%
Then there are functions $e_n(\sigma,y)$  such that, for all $n\in\N^*$ and environments $\alpha$ for which $[0,\sigma(K_n-1)]$ is a good block, the following bound holds:
%
\be
\label{limit_time}
{\tau}_n^\alpha(\sigma,y)\leq\tau^{\overline{\rho}_c,r/4}(\sigma,y)+e_n(\sigma,y).
\ee
{\color{black} Furthermore $e_n(\sigma,.)$ does not depend on $\alpha$ and
converges locally uniformly to 0 on $[\sigma^-,+\infty)$ as $n\to+\infty$.}
\end{corollary}
%
%


\begin{proof}[Proof of Corollary \ref{cor_block}]
%
The unrestricted passage time $T^\alpha(\sigma K_n,\lfloor K_n y\rfloor)$ 
may use paths that do not stay in $B := [0,\sigma K_n]$. 
To control the contribution outside $B$, we use a decomposition of the path in the same spirit as Section \ref{subsec:outline}. The problem here is simpler because there is no more
renormalization, and there are only three regions to consider for the path according to its $x$-coordinate (recall the simplifying notational convention $[a,b]=[b,a]$), 
namely the interval 
$[0,\sigma (K_n-1)]$ and the two intervals on either side of it, which are also bounded by the fact that the only  possible increments are $(1,0)$ and $(-1,1)$.
If $\sigma=1$, these intervals are $[-\lfloor K_n y\rfloor ,-1]$ and $[K_n, K_n + \lfloor K_n y\rfloor ]$. 
If $\sigma=-1$ then $y \geq 1$ and these intervals are 
$[-  \lfloor K_n y\rfloor  ,-K_n]$ and $[1,-K_n+ \lfloor K_n y\rfloor ]$.
%
We thus define a simpler path skeleton 
$(\tilde{z}_1,\tilde{y},\tilde{z}_2)=\tilde{\gamma}$ as described below. 

\medskip

Let $\gamma=(x_k,y_k)_{k=0,\ldots,m-1}$ be a path connecting
$(0,0)=(x_0,y_0)$ to $(x_{m-1},y_{m-1})=(\sigma K_n,[K_ny]=y')$. We set
%
\begin{eqnarray*}
%
& k_1 :=  1+
\max\{k=0,\ldots,m-1:\,\sigma x_k<0\},\\
%
& k_2 :=  \min\{k= k_1, \ldots,  m-1:\,x_k=\sigma K_n\},
%
\end{eqnarray*}
%
with the convention that the $\max$ is $-1$ if the corresponding set is empty. Since allowed  path increments are 
$(1,0)$ and $(-1,1)$, we have $x_{k_1}=0$. We then define (see figure \ref{fig: unrestricted})
%
\begin{eqnarray*}
%
\tilde{z}_1 :=  y_{k_1}, \qquad
\tilde{y}  :=  y_{k_2}-y_{k_1}, \qquad 
\tilde{z}_2 :=  y'-y_{k_2}.
\end{eqnarray*}
%
Let $\tilde{\Gamma}_n$  denote the set of these new ``skeletons'', that is the set of
triples $\tilde{\gamma}=(\tilde{z}_1,\tilde{y},\tilde{z}_2)$ such that 
%
\be\label{def_new_skeleton}
%
\tilde{z}_1+\tilde{y}+\tilde{z}_2=\lfloor K_n y\rfloor =:y',\quad
%
(\tilde{z}_1,\tilde{y},\tilde{z}_2)\in\N\times(\N\cap[\sigma^-K_n,+\infty)) \times \N.
%
\ee
The path between $k_1$ and $k_2$ corresponds to the restricted part which has already been studied in the previous sections. 
As in \eqref{double_max}, we write
%
\begin{eqnarray}
T^\alpha \big( (0,0), (K_n,\lfloor K_n y\rfloor ) \big)  = 
\max_{(\tilde{z}_1,\tilde{y},\tilde{z}_2)\in\tilde{\Gamma}_n }\left[
%
V^\alpha_1 (\sigma,\tilde{\gamma})+U^\alpha_B(\sigma,\tilde{\gamma})+V^\alpha_2(\sigma,\tilde{\gamma})
%
\right],
%
\label{cut_in_three}
%
\end{eqnarray}
%
where $B:=[0, \sigma K_n-1]$, and
%
\begin{eqnarray*}
%
V^\alpha_1(\sigma,\tilde{\gamma}) & := & T^\alpha((0,0),(0,\tilde{z}_1)),\\
%
U^\alpha_B(\sigma,\tilde{\gamma}) & := & T^\alpha_B\left(\left(\sigma,\tilde{z}_1+\frac{1-\sigma}{2}\right),\left(\sigma(K_n-1),\tilde{z}_1+\tilde{y}-\frac{1-\sigma}{2}\right)\right),\\
%
V^\alpha_2(\sigma,\tilde{\gamma}) & := & T^\alpha((\sigma K_n,\tilde{z}_1+\tilde{y}),(\sigma K_n,\tilde{z}_1+\tilde{y}+\tilde{z}_2)).
%
\end{eqnarray*}
%
%
%
%
Note that the second passage time in \eqref{cut_in_three} is restricted to $B$ by definition of the skeleton.
We then proceed as in Section \ref{subsec:outline} by studying the mean optimization problem (that is the maximum of the expectations of the three terms in \eqref{cut_in_three})
and estimating the error due to  this approximation.

\medskip

By definition of restricted passage times $\tau^\alpha_B$ and superadditivity bounds for vertical passage times,
%
\begin{eqnarray*}
%
\Exp \Big( V^\alpha_1(\sigma,\tilde{\gamma}) \Big) & \leq & \frac{4}{r}\tilde{z}_1,
\qquad \Exp \Big(  V^\alpha_2(\sigma,\tilde{\gamma}) \Big)  \leq  \frac{4}{r}\tilde{z}_2, \\
%
\Exp \Big(  U^\alpha_B(\sigma,\tilde{\gamma}) \Big)  & \leq  & K_n\tau^\alpha_B(\sigma,K_{n}^{-1}\tilde{y})\leq K_n\tau^{\rho_n,J_n}(\sigma,K_n^{-1}\tilde{y}),
\end{eqnarray*}
%
where the last inequality follows from Propositions \ref{prop:induction_gen} and \ref{prop:estimation_gen}.
%
From the definition  \eqref{ref_time} of $\tau^{\rho,J}$, we get for $i = 1,2$
$$
\frac{4}{r}\tilde{z}_i = \tau^{\overline{\rho}_c,r/4}(0,\tilde{z}_i)
\quad \text{and} \quad
\frac{1}{J_n}\tilde{z}_i = \tau^{\rho_n,J_n} (0,\tilde{z}_i).
$$
Thus we deduce that 
\begin{align}
& \frac{1}{K_n}
\left(
\frac{4}{r}\tilde{z}_1+\frac{4}{r}\tilde{z}_2+K_n\tau^{\rho_n,J_n}(\sigma,K_n^{-1}\tilde{y})
\right)
 \leq  \tau^{\overline{\rho}_c,r/4}(\sigma,y)\nonumber\\
%
& \qquad \qquad  \qquad \qquad  \qquad 
+  \tau^{\rho_n ,J_n}(\sigma,y)-\tau^{\overline{\rho}_c,r/4}(\sigma,y)
 +  \left|
%
\frac{4}{r} -  \frac{1}{J_n}
%
\right|y,
\label{upper_bound_three}
\end{align}
where we used that $ \tilde{z}_1 + \tilde{z}_2 \leq K_n y$.
%
Define $e_n^{(1)}(\sigma,y)$ as the second line
%the sum of the second and third terms 
of the r.h.s. of \eqref{upper_bound_three}. We have thus shown that
%
$$
\frac{1}{K_n} \max_{(\tilde{z}_1,\tilde{y},\tilde{z}_2)\in\tilde{\Gamma}_n }
\Big\{ 
%
\Exp \big( V^\alpha_1(\sigma,\tilde{\gamma}) \big) +
\Exp  \big(  U^\alpha_B(\sigma,\tilde{\gamma}) \big) +
\Exp  \big( V^\alpha_2(\sigma,\tilde{\gamma})  \big)
%
\Big\}
\leq \tau^{\overline{\rho}_c,r/4}( \sigma,y)+e_n^{(1)}(\sigma,y),
$$
where by the definition of $\tau^{\rho,J}$ \eqref{ref_time}, of $\overline{\rho}_c$ \eqref{eq: rhoc} and the convergence of $J_n$ to $r/4$, we deduce the  (locally uniform) convergence 
\margincom{{\color{red} mettre $e_n^{(1)}$ dans l'enonce ?}\textcolor{black}{pourquoi ?}}
$$
\lim_{n \to \infty} e_n^{(1)}(\sigma,.)  = 0.
$$
We conclude by controlling the error thanks to Proposition \ref{prop:fluct_2}, in the same spirit as Proposition \ref{prop:fluct}. 
\end{proof}
%
\begin{proposition}
\label{prop:fluct_2}
For  $\sigma\in\{-1,1\}$, there exist functions $e_n^{(2)}(\sigma,y)$ such that
\begin{align}
&  \Exp \left( \frac{1}{K_n} \; \max_{(\tilde{z}_1,\tilde{y},\tilde{z}_2)\in\tilde{\Gamma}_n }
\Big\{
V^\alpha_1(\sigma,\tilde{\gamma})
+U^\alpha_B(\sigma,\tilde{\gamma})
+V^\alpha_2(\sigma,\tilde{\gamma})
\Big\}
\right)
 \nonumber\\
%
& \qquad  -  \max_{(\tilde{z}_1,\tilde{y},\tilde{z}_2) \in \tilde{\Gamma}_n }
\Big\{ \frac{1}{K_n}
\Big[
%
\Exp \big( V^\alpha_1(\sigma,\tilde{\gamma}) \big)+
\Exp  \big(  U^\alpha_B(\sigma,\tilde{\gamma}) \big)+
\Exp  \big(  V^\alpha_2(\sigma,\tilde{\gamma})  \big)
%
\Big]  \Big\}
\leq  e_n^{(2)}(\sigma,y),
\label{fluct_2}
\end{align} 
and  $e_n^{(2)}(\sigma,.)$ converges locally uniformly to $0$ on $[\sigma^-,+\infty)$ as $n\to+\infty$.
%
\end{proposition}
%
\begin{proof}
%
We proceed in three steps as in the proof of Proposition \ref{prop:fluct}.\\ \\
%
{\bf Step 1.} {\it Cutoff.}
We again use a truncation procedure for the service times $Y_{i,j}$ as in step one of Subsection \ref{fluct_ent}.
%
%
 Here, we define  $\Prob'_{n,M,y'}$ as
 the distribution of the family
$(Y_{i,j}:\,i\in \textcolor{black}{[-  y', \sigma K_n+  y']},\; j\in[0,y'])$ conditioned on their maximum $M_B(y)$ being $rM$, and $\Prob_{n,M,y'}$ as the distribution of the family $(X_{i,j}:\,i\in \textcolor{black}{[-  \sigma y', \sigma K_n+  y']},\;  j\in[0,y'])$,
where $X_{i,j}$ are i.i.d. random variables, and the law of $X_{i,j}$ is the law of $Y_{i,j}$ conditioned on 
$Y_{i,j}\leq rM$. 
For  simplicity, we will only write $\Prob_M$ and $\Prob'_M$ for these distributions. 

\medskip

\noindent
{\bf Step 2.} {\it Fluctuations under cutoff.}
%
 Applying Lemma \ref{lemma_concentration} under $\Prob_{n,M,y'}$, we have
 %
 \begin{eqnarray}
 %
 V^\alpha_1(\sigma,\tilde{\gamma}) & \leq & 
 \Exp_M \big( V^\alpha_l(\sigma,\tilde{\gamma}) \big)
 +8M\sqrt{2\tilde{z}_1}Z_1  ,  \nonumber\\
 %
 V^\alpha_2(\sigma,\tilde{\gamma})
 %
  & \leq & \Exp_M \big(  V^\alpha_2(\sigma,\tilde{\gamma}) \big) + 8M\sqrt{2\tilde{z}_2}Z_2, 
  \nonumber\\
 %
 U^\alpha_B(\sigma,\tilde{\gamma}) & \leq & \Exp_M \big( U^\alpha_B(\sigma,\tilde{\gamma}) \big)
 %
 +8M\sqrt{ \sigma K_n+2\tilde{y}}Z_0,
 \label{concentration_three}
 \end{eqnarray}
 %
 where $Z_1$, $Z_2$ and $Z_0$ are independent random variables such that
 %
 $$
 \Prob_{n,M,y}(Z_k\geq t)\leq e^{-t^2},
 $$
for $k\in\{0,1,2\}$. 
We now apply Lemma \ref{cor_max} with $\mathcal A=\tilde{\Gamma}_n$, $\mathcal I=\{1,2,B\}$, 
and for $a=\tilde{\gamma}\in\tilde{\Gamma}_n$, ${\mathcal Y}_{a,1}=V^\alpha_1(\sigma,\tilde{\gamma})$,
${\mathcal Y}_{a,2}=V^\alpha_2(\sigma,\tilde{\gamma})$, ${\mathcal Y}_{a,B}=U^\alpha_B(\sigma,\tilde{\gamma})$,
$V_{a,1}=2\tilde{z}_1$, $V_{a,2}=2\tilde{z}_2$, $V_{a,B}=\sigma K_n+2\tilde{y}$.
Since (see \eqref{def_new_skeleton})
%
%
\be\label{nb_new_skeletons}
%
|\tilde{\Gamma}_n|=\left(
%
\ba{l}
%
2+ \lfloor K_n y\rfloor-\sigma^- K_n\\
%
2
%
\ea
%
\right)
\leq K_n^2 (1+y)^2,
%
\ee
%
%Thus Lemma \ref{cor_max} yields 
%
we obtain
\begin{align}
& 
 \Exp_M \left( \frac{1}{K_n} \; \max_{(\tilde{z}_1,\tilde{y},\tilde{z}_2)\in\tilde{\Gamma}_n }
\Big\{
V^\alpha_1(\sigma,\tilde{\gamma})
+U^\alpha_B(\sigma,\tilde{\gamma})
+V^\alpha_2(\sigma,\tilde{\gamma})
\Big\}
\right)
 \nonumber\\
%
&\qquad \quad \leq \max_{(\tilde{z}_1,\tilde{y},\tilde{z}_2) \in \tilde{\Gamma}_n }
\Big\{ \frac{1}{K_n}\left[
%
\Exp_M  \big(  V^\alpha_1(\sigma,\tilde{\gamma}) \big) +
\Exp_M \big(  U^\alpha_B(\sigma,\tilde{\gamma}) \big) +
\Exp_M \big(  V^\alpha_2(\sigma,\tilde{\gamma}) \big)
%
\right]  \Big\}
%
\label{fluct_3}\\
&  \qquad \qquad \qquad  \nonumber 
+
8M \frac{ \sqrt{\sigma+2y}}{\sqrt{K_n}} 
\left(\sqrt{3\pi}+
%\sqrt{\pi}
\sqrt{\pi}\sqrt{A}+\sqrt{A}\sqrt{\log|\tilde{\Gamma}_n|}
\right)\nonumber\\
&\qquad \quad \nonumber 
\leq \max_{(\tilde{z}_1,\tilde{y},\tilde{z}_2) \in \tilde{\Gamma}_n }
\Big\{ \frac{1}{K_n}\left[
%
\Exp  \big( V^\alpha_1(\sigma,\tilde{\gamma}) \big) +
\Exp \big( U^\alpha_B(\sigma,\tilde{\gamma}) \big) +
\Exp \big( V^\alpha_2(\sigma,\tilde{\gamma}) \big)
%
\right]  \Big\} \\
& \qquad \qquad \qquad 
+  8M
\frac{
%
\sqrt{\sigma+2y}
%
}{
%
\sqrt{K_n}
%
}\left(
%
\sqrt{3\pi}+
%\sqrt{\pi}
\sqrt{\pi}\sqrt{A}+\sqrt{A}\sqrt{\log|\tilde{\Gamma}_n|}
%
\right).
\nonumber
%
\end{align} 
%
In the last inequality, we have used the fact that the passage times under $\Prob_M$ are stochastically dominated by the passage times under $\Prob$.\\ \\
%
{\bf Step 3.} {\it Removing the cutoff.}
%
As in step three of the proof of Lemma \ref{cor_max}, a coupling argument shows that
the distribution under $\Prob'_{M}$ of any passage time $T$ depending only on the previous set of $Y_{i,j}$ is dominated by the distribution under $\Prob_{M}$ of $T+M$. Therefore 
%
\begin{eqnarray}
& & \Exp'_M \left( \frac{1}{K_n} \; \max_{(\tilde{z}_1,\tilde{y},\tilde{z}_2)\in\tilde{\Gamma}_n }
\Big\{
V^\alpha_1(\sigma,\tilde{\gamma})
+U^\alpha_B(\sigma,\tilde{\gamma})
+V^\alpha_2(\sigma,\tilde{\gamma})
\Big\}
\right)\nonumber\\
& &\qquad  \quad \nonumber 
\leq \max_{(\tilde{z}_1,\tilde{y},\tilde{z}_2) \in \tilde{\Gamma}_n }
\Big\{ \frac{1}{K_n} \Big(
%
\Exp \big(  V^\alpha_1(\sigma,\tilde{\gamma}) \big) +
\Exp \big( U^\alpha_B(\sigma,\tilde{\gamma}) \big) +
\Exp \big( V^\alpha_2(\sigma,\tilde{\gamma}) \big)
%
\Big)  \Big\} \\
%
%
&  &\qquad \qquad+ \frac{M}{K_n}+ 8M\frac{
%
\sqrt{\sigma+2y}
%
}{
%
\sqrt{K_n}
%
}\left(
%
\sqrt{3\pi}+
%\sqrt{\pi}
\sqrt{\pi}\sqrt{A}+\sqrt{A}\sqrt{\log|\tilde{\Gamma}_n|}
%
\right).
\nonumber
%
\end{eqnarray} 
%
Integrating the above inequality with respect to the distribution of $M_B(y)$ yields \eqref{fluct_2}, with
%
$$
e_n^{(2)}(\sigma,y)  :=  m \big(  \lfloor K_n y\rfloor (\sigma K_n+2 \lfloor K_n y\rfloor) \big) \, E_n(\sigma,y),
$$
%
where $m(.)$ satisfies the bound \eqref{bound_expmax}, and
%
$$
E_n(\sigma,y):= \frac{1}{K_n}+ 8\frac{
%
\sqrt{\sigma+2y}
%
}{
%
\sqrt{K_n}
%
}\left(
%
\sqrt{3\pi}+
%\sqrt{\pi}
\sqrt{\pi}\sqrt{A}+\sqrt{A}\sqrt{\log|\tilde{\Gamma}_n|}
%
\right),
$$
%
from which one can see that $e_n^{(2)}(\sigma,.)$ converges locally uniformly to $0$.
%
%
\end{proof}





\subsection{Proof of Theorems \ref{thm: passage upper} and \ref{th_plateau}}
\label{conclude}
%
%
Theorem  \ref{th_plateau} is a consequence of Theorem \ref{thm: passage upper} which we prove now.
Given $\sigma\in\{-1,1\}$, by Theorem \ref{th_sep}, we have 
$$
\tau_\varepsilon(\sigma,y)  =  \lim_{n\to\infty} \mathcal E_\varepsilon \times \Exp
\left( \frac{1}{K_n}T(\sigma K_n, \lfloor K_n y\rfloor ) \right)
= \lim_{n\to\infty}\mathcal E_\varepsilon\left(\tau^\alpha_n(\sigma,y)\right), 
$$
where $\mathcal E_\varepsilon$ stands for the expectation with respect to the disorder
$\alpha$.
%
Note that the above limit does not follow directly from Theorem \ref{th_sep}, which yields an a.s. limit. 
However, the convergence in Theorem \ref{th_sep} holds also in $L^1$. This follows from a quasi-Gaussian tail estimate for the passage time $T(\sigma K_n, \lfloor K_n y\rfloor)$, obtained from Lemma \ref{lemma_concentration} and a cutoff as in step four of Subsection \ref{fluct_ent}.
%
%\sout{Let us denote by ${\Exp}_\varepsilon$  the expectation with respect to the product measure $\mathcal P_\varepsilon\otimes\Prob$, that is the joint expectation with respect to disorder
%$\alpha$ and service times $(Y_{i,j})$.} Let $\sigma\in\{-1,1\}$, and 
Let ${\bf G}_n(\sigma)$ be the set of environments $\alpha$ for which $[0,\sigma(K_n-1)]$ is a good block. 
The mean passage time 
%$\tau_\varepsilon(\sigma,y)$ 
can be decomposed as
$$
\mathcal E_\varepsilon[\tau^\alpha_n(\sigma,y)]
=
\mathcal E_\varepsilon\left[
%
\tau^\alpha_n(\sigma,y)\indicator{{\bf G}_n(\sigma)}
%
\right]+\mathcal E_\varepsilon\left[
%
\tau^\alpha_n(\sigma,y)\indicator{{\bf A}\backslash{\bf G}_n(\sigma)}
%
\right],
$$
%
where ${\bf A}:=[0,1]^\Z$ is the set of environments.
By Corollary \ref{cor_block} (recall that the function \textcolor{black}{$e_n$} in \eqref{limit_time} does not depend on $\alpha$), the $\limsup$ of the first term is bounded above by $\tau^{\overline{\rho}_c,r/4}( \sigma,y)$. On the other hand, the second term is bounded above by
%
$$
%
\mathcal E_\varepsilon\left[ \left(\tau^\alpha_n(\sigma,y)\right)^2\right]^{1/2}
\,
\mathcal P_\varepsilon
%
\left[
%
{\bf A}
\backslash
{\bf G}_n(\sigma)
\right]^{1/2}.
$$
%
The $\mathcal P_\varepsilon$-probability vanishes as $n\to\infty$ by Lemma \ref{lemma_good}, while
the expectation of the squared passage time {\color{black} can be bounded by} 
$$
\mathcal E_\varepsilon \left[ \left(\tau^\alpha_n(\sigma,y)\right)^2\right]
\leq\tau_n(\sigma,y)^2,
$$
%
where $\tau_n(\sigma,y)$ is defined as \eqref{def_unrestricted} for a homogeneous environment $\alpha(x)\equiv r$ (that is for rate $r$ homogeneous TASEP).
%
The limit $\tau_n(\sigma,y)\to r^{-1} (\sqrt{\sigma+y}+\sqrt{y})^2$ as $n \to \infty$,  follows from the above remark on $L^1$-convergence of rescaled passage times in Theorem \ref{th_sep}. This implies
$\tau_n(\sigma,y)^2\to r^{-2} (\sqrt{\sigma+y}+\sqrt{y})^4$ as $n\to\infty$. 
We finally get
\be
\label{final_bound_passage}
\tau_\varepsilon({\color{black} \sigma},y)\leq\tau^{\overline{\rho}_c,r/4}( {\color{black} \sigma} ,y),
\ee
%
for every ${\color{black} \sigma} \in\{-1,1\}$ and $y \geq {\color{black} \sigma}^-$. Since $\tau_\varepsilon$ and $\tau^{\overline{\rho}_c,r/4}$ are homogeneous functions, 
\eqref{lower_bounds} follows  for $\rho=\overline{\rho}_c$.


We now show that
%
$$
\lim_{\varepsilon\to 0}\rho_c(\varepsilon)=\rho_c(0).
$$
%
Indeed, by \eqref{critical_density_2} and \eqref{final_bound_passage},
we have $\rho_c\leq\overline{\rho}_c$.
Then, by Proposition \ref{prop: estimation suites}, 
%
$$
\limsup_{\varepsilon\to 0}{\rho}_c(\varepsilon)
\leq
\limsup_{\varepsilon\to 0}\overline{\rho}_c(\varepsilon)\leq\rho_c(0).
$$
{\color{black} The reversed inequality will be proved by contradiction. Suppose that we have}
$$
\liminf_{\varepsilon\to 0}\rho_c(\varepsilon)<\rho_c(0),
$$
%
then for some $\varepsilon>0$ we would have $\rho_c(\varepsilon)<\rho_c(0)$, hence
%
$$
\frac{r}{4}=\max f_\varepsilon=f_\varepsilon[\rho_c(\varepsilon)]
\leq f_{\rm TASEP}[\rho_c(\varepsilon)]<f_{\rm TASEP}[\rho_c(0)]=\frac{r}{4},
$$
%
where $f_{\rm TASEP}$ denotes the flux of the homogeneous rate 1 TASEP, and the last inequality follows from $\rho_c(\varepsilon)<\rho_c(0)\leq 1/2$. 






\subsection{Proof of Theorems \ref{th:passage_dilute} and \ref{th_dilute_limit}}
\label{sec:proof_dilute}

{\color{black} 
Using Proposition \ref{prop:passage_dilute}, we are going to derive the limiting passage time of 
Theorem \ref{th:passage_dilute} and then conclude Theorem \ref{th_dilute_limit}.


%In this subsection, we derive Theorem \ref{th:passage_dilute}, and thus Theorem \ref{th_dilute_limit}, from Proposition \ref{prop:passage_dilute}.

\medskip

By coupling with a rate $1$ homogenous TASEP, we get 
\begin{equation}
\label{trivial_bound}
\tau_\varepsilon(1,y)\ge g_1(1,y).
\end{equation}
Combining this with Proposition \ref{prop:passage_dilute}, 
we deduce that 
}
$$
\forall y<y_1, \qquad 
\lim_{\varepsilon\to 0}\tau_\varepsilon(1,y)= g_1(1,y) = 
(\sqrt{1+y}+\sqrt{y})^2,
$$
where $\tau_\varepsilon$ denotes the limiting rescaled passage time. 
%We have thus shown
%$$
%\lim_{\varepsilon\to 0}\tau_\varepsilon(1,y)=g_1(1,y)=(\sqrt{1+y}+\sqrt{y})^2,\quad\forall y<y_1
%$$
%
%
Similarly, one can show that 
%
$$
{\color{black} \forall  y \in [1,y'_1]}, \qquad 
\lim_{\varepsilon\to 0}\tau_\varepsilon(-1,y)=g_1(-1,y)=(\sqrt{-1+y}+\sqrt{{y}})^2.
$$
%
As the height profile $h_\varepsilon(t,x)=tk_{\textcolor{white}{\varepsilon}}(\frac{x}{t})$ \eqref{corr_4}
is the inverse of $\tau_\varepsilon(x,y)$ wrt $y$, we obtain
%%
\begin{equation}\label{k_tasep}
\lim_{\varepsilon\to 0}k_\varepsilon(v)=\frac{(1-v)^2}{4},\quad\forall v\in [1-2\rho_1(0),1]\cup [-1,2\rho_1(0)-1].
\end{equation}
%
Next we use
%
\begin{equation}\label{duality}
f_\varepsilon(\rho)=\inf_{v}[\rho v+k_\varepsilon(v)].
\end{equation}
%
For $\rho\not\in[\rho_1(0),1-\rho_1(0)]$, it follows from (\ref{k_tasep}) that the minimum in (\ref{duality}) is achieved for
$v=v_\varepsilon\to 1-2\rho$ as $\varepsilon\to 0$, thus
%
$$
\lim_{\varepsilon\to 0}f_\varepsilon(\rho)=\rho(1-\rho).
$$
%
Since the above expression takes value $r/4$ for $\rho\in\{\rho_1(0),1-\rho_1(0)\}$, 
and $f_\varepsilon$ is a concave function with maximum value $r/4$, 
we then necessarily have
%
$$
\lim_{\varepsilon\to 0}f_\varepsilon(\rho)=\frac{r}{4},\quad\forall \rho\in[\rho_1(0),1-\rho_1(0)].
$$





\begin{appendix}
%
%\section{Equivalence of flux definitions}
%\label{app:equivalence}
%%
%In this section, we show the validity of Definition \eqref{flux_current} 
%\textcolor{black}{of the flux}
%(that is existence of the limit and independence with respect to $x$ and $\alpha$) and its equivalence with the alternative definitions \eqref{flux_torus} and \eqref{flux_line}. Let us first give details of the latter definitions.\\ \\
%%
%%For fixed $L$, the expectation in \eqref{flux_torus} is well-defined because the system is a finite state irreducible jump Markov process, hence it has a unique invariant measure
%%$\nu^\alpha_{N,L}$. The issue is thus to estalish convergence as $L\to\infty$ to the same limit as given by \eqref{flux_current}.
%%
%{\em Stationary current on the torus.}
%%
%%Considering the ring has the advantage that existence and uniqueness for the equilibrium state with given particle number is straightforward. However the existence
%%of a thermodynamic limit is not trivial in general. 
%%
%Let $\mathbb{T}_N:=\{0,\ldots,N-1\}$ with periodic boundary conditions. We consider the 
%Markov process on $\{0,1\}^{\mathbb{T}_N}$ with generator  given by 
%%
%\be\label{generator_torus}
%%
%L^\alpha_N f(\eta)=\sum_{x\in\mathbb{T}_N}\alpha(x)\eta(x)[1-\eta(x+1)]\left[
%%
%f\left(\eta^{x,x+1}\right)-f(\eta)
%%
%\right]
%%
%\ee
%%
%Since $N$ is identified with $0$,  a particle at $N-1$ jumps to $0$. This defines a Markov jump process with finite state space. Due to conservation of particle number, each  set
%%
%\be\label{hyperplanes}
%%
%{\bf X}_{N,k}:=\left\{
%%
%\eta\in{\bf X}:\,\sum_{x\in\mathbb{T}_N}\eta(x)=k
%%
%\right\}
%%
%\ee
%%
%is invariant. The restriction of the process to ${\bf X}_{N,k}$ is irreducible. Thus it has a unique invariant measure $\nu_{N,k}^\alpha$. 
%The expectation in \eqref{flux_torus} is well defined for all $N\in\N^*$ and $k\in\N$, but the issue is to show the thermodynamic limit.
%%
%We may alternatively view the dynamics defined by \eqref{generator_torus} as a Markov process on the subset
%%
%$$
%{\bf X}_N^{\rm per}:=\{\eta\in{\bf X}:\,\forall x\in\Z, \qquad \eta(x+N)=\eta(x)\}
%$$
%%
%of periodic particle configurations on $\Z$ with generator
%%
%\be\label{generator_torus_2}
%%
%L^\alpha_N f(\eta)=\sum_{x=0}^{N-1}\alpha(x)\eta(x)[1-\eta(x+1)]\left[
%%
%f\left(\eta^{x,x+1,{\rm per}}\right)-f(\eta)
%%
%\right]
%%
%\ee
%%
%where $\eta^{x,x+1,{\rm per}}$ is the periodic configuration obtained by letting particles jump from $x+kN$ to $x+1+kN$ for all $k\in\Z$. The process defined by  \eqref{generator_torus_2}
 %is equivalent to \eqref{generator_torus} in the sense that  the restriction to $\{0,\ldots,N-1\}$ is a Markov process with generator \eqref{generator_torus}. Conversely, the periodic extension
%of a process generated by \eqref{generator_torus} is a process generated by \eqref{generator_torus_2}.\\ \\
%%
%{\em Stationary current on the line.}
%%
%We consider now the system on $\Z$ with generator \eqref{generator}.
%Let $\mathcal R$ be the set of densities $\rho\in[0,1]$ with the following property: there exists a subset ${\bf B}\subset{\bf A}$ {\color{black} of the jump rate configurations}, and a family of probability measures $(\nu^\alpha_\rho)_{\alpha\in{\bf B}}$, such that:\\ \\
%%
%(o) $\tau_1 {\bf B}={\bf B}$ a.s., and $\bf B$ has probability $1$ with respect to the law of the environment.\\ \\
%%
%(i) For every $\alpha\in{\bf B}$, $\nu_\rho^\alpha$ is invariant for $L^\alpha$.\\ \\
%%
%(ii) $\nu_\rho^\alpha$ has density $\rho$ in the sense that
%%
%\be\label{density_invariant}
%%
%\lim_{l\to\infty}\frac{1}{2l+1}\sum_{x\in\Z:\,|x|\leq l}\eta(x)=\rho,\quad
%\nu_\rho^\alpha \mbox{-a.s.}
%%
%\ee
%%
%(iii) For a.e. $\alpha\in{\bf B}$ and $x\in\Z$, \ \  $\nu_\rho^{\tau_x\alpha}=\tau_x 
%\nu_\rho^\alpha$\\ \\
%%
%Note that $\mathcal R$ clearly contains $0$ and $1$, since the deterministic configurations with either $0$'s or $1$'s everywhere are invariant for any environment $\alpha$.
%The following is shown in \cite{bgrs4}: (1) $\mathcal R$ is a closed subset of $[0,1]$; (2) the quantity $f(\rho)$ defined by
%%
%%
%\be\label{flux_line_precise}
%%
%f(\rho):=\left\{
%%
%\ba{lll} 
%%
%\dsp \int j_x^\alpha(\eta)d\nu^\alpha_\rho(\eta) & \mbox{if} & \rho\in\mathcal R\\
%%
%\dsp\frac{\rho^+-\rho}{\rho^+-\rho^-}f(\rho^-)+\frac{\rho-\rho^-}{\rho^+-\rho^-}f(\rho^+)& \mbox{if} & \rho\not\in\mathcal R
%%
%\ea
%%
%\right.
%%
%\ee
%%
%where
%%
%\be\label{rhopm}
%%
%\rho^-=\sup\mathcal R\cap[0,\rho)
%%
%,\quad
%%
%\rho^-=\inf\mathcal R\cap(\rho,1]
%%
%\ee
%%
%does not depend on the disorder realization $\alpha$; (3) the system has a quenched hydrodynamic limit given by a scalar conservation law with flux function $f(\rho)$, see Theorem \ref{hdl_disorder} below for a precise statement.
%An easy observation (see \cite{sep}) is that
%$\mathcal R$ and $f$ are symmetric if the distribution of $\alpha$ is invariant by symmetry $x\mapsto -x$.
%%
%%
%%
%\textcolor{black}{
%From now, this appendix is  devoted to the proof of the following proposition.}
%%
%\begin{proposition}
%\label{prop:flux}
%%
%For $\mathcal P$-a.e. $\alpha\in{\bf A}$,
%the limits in \eqref{flux_current} and  \eqref{flux_torus} exist for all $x\in\Z$ and $\rho\in\mathcal R$. These limits do not depend on $x$ or $\alpha$, and coincide with the quantity defined by \eqref{flux_line_precise}--\eqref{rhopm} 
%%
%\end{proposition}
%%
%The main ingredients for this proof are the hydrodynamic limit for site-disordered TASEP on the line and torus, stated in Subsection \ref{subsec_hdl} below, and a result on the asymptotic flux
%in \eqref{conservation_law} on the torus, stated in Subsection \ref{subsec_asymtotic_flux} below.\\
%%
%\begin{remark}\label{remark_set_r}
%%
%Nothing is known about the above set $\mathcal R$. In particular, whether it is a strict subset of $[0,1]$, and whether it contains values other than $0$ and $1$,
%are open questions. A consequence of Theorem \ref{th_plateau} and Proposition \ref{prop:flux}  is that, under assumption \hh, $\mathcal R$ contains at least two other values.\margincom{{\bf CH} vaut peut etre le coup d'en faire un corollaire}
%%
%\end{remark}
%%
%%{\bf Ceci est a replacer au bon endroit, aussi pour le tore.}
%%Note that the above expectation does not depend on $x$, because $\nu_{N,k_N}$ is stationary, and
%%%
%%\be\label{indep_x}
%%j^\alpha_{x-1}(\eta)-j^\alpha_{x}(\eta)=L^\alpha[\eta(x)]
%%\ee
%%%
%%%
%%The existence of the limits \eqref{flux_current} and \eqref{flux_current_3}, and its identification with the
%%right-hand side of \eqref{flux_line}, will follow from hydrodynamic limit arguments.
%%%
%%%
%%Hydrodynamic limit was established in \cite{sep} for TASEP with i.i.d. site disorder, and more generally in \cite{bgrs4}, for attractive systems in ergodic random environment.
%%We first recall this result, and some consequences that will be used for the proof of equivalence.
%%
%\subsection{Hydrodynamic limit of site-disordered TASEP}\label{subsec_hdl}
%%
%Let $\mathcal M(\R)$ denote the set of Radon measures on $\R$ equipped with the topology of vague convergence.
%Define 
%%
%\be\label{def_empirical}
%%
%\pi^N(\eta)(dx):=N^{-1}\sum_{y\in\Z}\eta(y)\delta_{y/N}(dx)\in\mathcal M(\R)
%%
%\ee
%%
%as the empirical density field viewed at scale $N$ under the particle configuration $\eta$. Let  ($\eta^N)_{N\in\N^*}$ be a sequence of $\bf X$-valued random variables, and 
%$\rho(.)$ be an $L^\infty(\R;[0,1])$-valued random field. We write $\eta^N\sim\rho(.)$, iff. the sequence of random measures $\pi^N(\eta^N)(dx)$ converges in distribution
%to $\rho(.)dx$. In the periodic setting, when working on ${\bf X}_N^{{\rm per}}$ with generator \eqref{generator_torus_2}, $\rho(.)$ is a.s. a $1$-periodic function.
%%
%%When working on the torus, using description \eqref{generator_torus_2}, the empirical measure is defined instead of \eqref{def_empirical} by
%%
%%\be\label{def_empirical_torus}
%%
%%\pi^N(\eta)(dx):=N^{-1}\sum_{y=0}^{N-1}\eta(y)\delta_{y/N}(dx)
%%\in\mathcal M(\R)
%%
%%\ee
%%
%%We can similarly define $\eta^N\sim\rho(.)$, when ($\eta^N)_{N\in\N^*}$ is a sequence of ${\bf X}^{N,{\rm per}}$-valued random variables and $\rho_0(.)$ an $L^\infty(\mathbb{T})$ random field.
%%
%The following result is an extension of the results of \cite{sep} and \cite{bgrs4} to random initial data and to the periodic setting \eqref{generator_torus_2}. 
%It will be proved in Subsection \ref{proof_hdl}.
%%
%%
%\begin{theorem}\label{hdl_disorder}
%%
%Let $f$ be defined by \eqref{flux_line_precise}--\eqref{rhopm}.
%Then for $\mathcal P$-a.e. environment $\alpha(.)\in\bf A$, the following holds.
%%
%Let $(\eta^N_t)_{t\geq 0}$ be a  sequence of processes with generator \eqref{generator} or \eqref{generator_torus_2}. Assume the sequence of initial configurations $\eta^N_0$ is such that $\eta^N_0\sim\rho_0(.)$. Let $\Pi^N_.:=\pi^N_{N.}(\eta^N_{N.})(dx)$ denote the rescaled empirical measure process. Then\\ \\
%%
%(1) The sequence
%$(\Pi^N_.)_{N\in\N^*}$ converges in distribution in the Skrorokhod space $\mathcal D([0,\infty),\mathcal M(\R))$ 
%to the random path $\Pi_.$ defined by $\Pi_t=\rho_t(x)dx$,
%where $\rho_t(x)=\rho(t,x)$ is the unique entropy solution to the scalar conservation law
%%
%\be\label{conservation_law}
%%
%\partial_t\rho(t,x)+\partial_x f[\rho(t,x)]=0
%%
%\ee
%%
%%
%with Cauchy datum $\rho_0(.)$.
%This convergence also holds in the local uniform topology.
%In particular, for every $t>0$, $\eta^N_{Nt}\sim\rho_t(.)$. \\ \\
%%
%(2) The following scaling limit holds in distribution for the current for all $t\geq 0$, and either $x\in\R$ (in the case \eqref{generator}) or $x\in\mathbb{T}$ (in the case \eqref{generator_torus_2}):
%%
%\be\label{scaling_limit_current}
%%
%\lim_{N\to\infty}N^{-1}J^\alpha_{Nx}(Nt,\eta^N_{0})=\int_0^t f[\rho(s,x)]ds
%%
%\ee
%%
%%
%\end{theorem}
%%
%\subsection{Asymptotic current for periodic solutions of \eqref{conservation_law}}\label{subsec_asymtotic_flux}
%%
%In this subsection, we state and prove the following result
%%
%\begin{proposition}
%\label{prop_large_time}
%%
%Let $\rho_0(.)\in L^\infty(\R)$ be $1$-periodic and $[0,1]$-valued, with 
%%
%\be\label{rhobar}
%\bar{\rho}=\int_0^1\rho_0(x)dx
%\ee
%%
%Let $\rho(t,x)$ denote the entropy solution to \eqref{conservation_law} with Cauchy datum $\rho_0(.)$. Then, for every $x\in\R$,
%%
%\be\label{asymptotic_flux_entropy}
%%
%\lim_{t\to\infty}\frac{1}{t}\int_0^tf(\rho(s,x)) ds=f(\bar{\rho})
%%
%\ee
%%
%\end{proposition}
%%
%\begin{proof}
%%
%Let $h_0(x):=\dsp\int_0^x\rho_0(y)dy$ and $\tilde{h}_0(x)=\bar{\rho} x$. Since $\rho_0$ is $1$-periodic and satisfies \eqref{rhobar}, $h_0$ and 
%$\tilde{h}_0$ coincide on the integers, hence
%%
%\be\label{max_diff}
%%
%\sup_{x\in\R}|h_0(x)-\tilde{h}_0(x)|<+\infty
%%
%\ee
%%
%Let $h_t(x)$ and $\tilde{h}_t(x)$ denote the viscosity solutions of the Hamilton-Jacobi equation
%%
%\be\label{hj}
%\partial_t h+f(\partial_x h)=0
%\ee
%%
%with respective initial data $h_0$ and  $\tilde{h}_0$. It is well-known (\cite{bar}) that (i)
%$\partial_x h_t=:\rho_t$ is the entropy solution to \eqref{conservation_law} with initial datum $\partial_x h_0=\rho_0$, (ii)
%$\tilde{h}_t$ is given by
%%
%\be\label{viscosity_uniform}
%\tilde{h}_t(x)=\bar{\rho}x-tf(\bar{\rho}),
%\ee
%%
%and (iii) $h_t$ and $\tilde{h}_t$ satisfy the maximum principle
%%
%\be\label{max_principle}
%\sup_{x\in\R}|h_t(x)-\tilde{h}_t(x)|\leq\sup_{x\in\R}|h_0(x)-\tilde{h}_0(x)|
%\ee
%%
%It follows from \eqref{viscosity_uniform}, \eqref{max_principle} and \eqref{max_diff} that
%%
%$$
%\lim_{t\to\infty}\frac{1}{t}[h_t(x)-h_0(x)]=\lim_{t\to\infty}h_t(x)=\bar{\rho}
%{\color{black} ????}$$
%%
%But by \eqref{hj}, since $\rho_t=\partial_x h_t$, the l.h.s. of the above equality is equal to the l.h.s. of \eqref{asymptotic_flux_entropy}.
%%
%\end{proof}
%%
%\subsection{Proof of Proposition \ref{prop:flux}}
%%
%\mbox{}\\ \\
%%
%Let us first remark that \textcolor{black}{if the limit \eqref{flux_current} exists for $x=0$, then it exists 
%for any $x\in\N$ and is constant.}
%This is due to the relation
%%
%$$
%J_x^\alpha(t,\eta)-J_y^\alpha(t,\eta)=\sum_{z=x+1}^y\eta_0(z)-\sum_{z=x+1}^y\eta_t(z)
%$$
%%
%for any $x,y\in\Z$ with $x<y$, and the fact that $\eta_t(z)\in\{0,1\}$.\\ \\
%%
%{\em Step 1. Equivalence of \eqref{flux_current} and \eqref{flux_line_precise}--\eqref{rhopm}.} 
%In the setting of \eqref{flux_current}, 
%set $\eta^N_0=\eta^\rho$ for all $N\in\N^*$. 
%Then \eqref{uniform_profile} implies $\eta^N_0\sim\rho_0(.)$ 
%with $\rho_0(.)\equiv\rho$.
%Since the entropy solution of \eqref{conservation_law} is then given by $\rho(t,x)\equiv\rho$, taking $t=0$ and $x=0$ in \eqref{scaling_limit_current} establishes \eqref{flux_current} along integer times $t\to\infty$  for $x=0$.
%To relax the restriction on integer times, we note that the inequalities
%%
%$$
%J_x^\alpha(t,\eta)-J_x^\alpha([t],\eta)\leq N(t)-N([t])\leq X
%$$
%%
%hold in distribution, where $[t]$ denotes the integer part of $t$, $N(.)$ is a rate $1$ Poisson process, and $X$ a Poisson random variable with parameter $1$.\\ \\
%%
%%One uses the following standard fact: hydrodynamic limit implies that the scaling limit for the current is the time intergated flux for the entropy solution. Precisely, under assumptions
%%of Theorem \ref{hdl_disorder},
%%%
%%$$
%%N^{-1}J^\alpha_{Nx}(Nt,\eta^N_0)\to\int_0^t f[\rho(s,x)]ds
%%$$
%%%
%%{\bf A completer.}\\ \\
%%
%{\em Step 2. Equivalence of \eqref{flux_line_precise}--\eqref{rhopm} and \eqref{flux_torus}.} 
%%
%%We still denote by $\nu^\alpha_{N,k_N}$ the invariant measure of \eqref{generator_torus} under the point of view \eqref{generator_torus_2}, that is after periodic completion on $\Z$ of the random particle configuration. 
%%
%Let $(k_N)_{N\in\N^*}$ be a sequence of integers such that
%%
%\be\label{limiting_density}
%\lim_{N\to\infty}\frac{k_N}{N}=\rho
%\ee
%%
%For notational simplicity we write $\nu_N^\alpha$ for $\nu_{N,k_N}^\alpha$.
%By stationarity of the latter,
%%
%\be\label{stationary}
%%
%\int_{{\bf X}^{\rm per}_N}j_0(\eta)d\nu^\alpha_{N}(\eta)  =  \Exp_{{N}}^\alpha\left\{
%%
%\frac{1}{Nt}\int_0^{Nt} j_0^\alpha(\eta_s^\alpha)ds
%%
%\right\}
%%
%=\Exp_{N}^\alpha\left[\frac{1}{Nt}J^\alpha_0(Nt,\eta)\right]
%%
%\ee
%%
%where $\Exp_{N}^\alpha$ denotes the expectation w.r.t. the process with generator \eqref{generator_torus_2} and initial distribution $\nu_{N}^\alpha$. The law of this process will be denoted below by
%$\Prob_{N}^\alpha$.
%Let $Q^N=\nu^\alpha_{N,k_N}\circ(\pi^N)^{-1}$ denote the distribution of $\pi^N(\eta)$ when $\eta$ has distribution $\nu^\alpha_{N,k_N}$. Since $\mathcal M(\R)$ is compact for the topology of vague convergence, $(Q^N)_{N\in\N^*}$ is tight. 
%Let $Q$ be one of its limit points. In the following, $\lim_{N\to\infty}$ implicitely denotes convergence along the associated subsequence. 
%Since $\eta(x)\in\{0,1\}$ for all $x$, $Q$ is supported on Radon measures $\pi$ such that $\pi(I)\leq|I|$ for every interval $I$ of $\R$. 
%Hence we may view $Q$ as the distribution of a random $1$-periodic element $\rho_0(.)\in L^\infty(\R)$ with values in $[0,1]$. Besides, \eqref{limiting_density} implies
%that
%%
%\be\label{mean_density}\int_0^1\rho_0(x)dx=\rho\ee
%%
%with probability $1$.
%By Theorem \ref{hdl_disorder}, for a.e. disorder realization $\alpha$, the limit
%%
%\be\label{lim_cur_prob}
%%
%\frac{1}{Nt}J^\alpha_0(Nt,\eta)\stackrel{t\to\infty}{\longrightarrow}\Gamma_t:=\frac{1}{t}\int_0^t f[\rho(s,x)]ds
%%,\quad\mbox{in }\Prob_{\nu_{N,k_N}}\mbox{ probability}
%%
%\ee
%%
%holds in $\Prob_N^\alpha$-probability with respect to $\eta$,
%where $\rho(.,.)$ is the entropy solution to \eqref{conservation_law} with random Cauchy datum $\rho_0(.)$ (note that the randomness of $\Gamma_t$ comes only from the randomness of $\rho_0(.)$). 
%%
%Since $J_0^\alpha(t,\eta)$ is bounded above in distribution by a Poisson random variable with parameter $t$, the l.h.s. of \eqref{lim_cur_prob} is a uniformly integrable family of random variables, hence
%
%%
%$$\lim_{N\to\infty}\Exp_{N}^\alpha J^\alpha_0(Nt,\eta)=\Exp(\Gamma_t)$$
%%
%%
%By Proposition \ref{prop_large_time}, $1$-periodicity of $\rho_0(.)$ and \eqref{mean_density}, $\Gamma_t\to f(\rho)$ in probability as $t\to\infty$.
%Since $f$ is uniformly bounded, $\Gamma_t$ is uniformly bounded, thus
%%
%$$
%\lim_{t\to\infty}\Exp\Gamma_t=f(\rho)
%$$
%%
%Finally, since we may choose $t$ arbitrarily large in \eqref{stationary},  the result follows.
%%
%%
%\subsection{Proof of Theorem \ref{hdl_disorder}}\label{proof_hdl}
%%
%For this proof, we recall some standard material.\\ \\
%%
%\textbf{Entropy inequalities.} The entropy solution $\rho(t,x)$ to \eqref{conservation_law} with Cauchy datum $\rho_0(.)$ is uniquely characterized (see e.g. \cite{kru})  by the fact that, for every $c\in[0,1]$, and every 
%$\varphi\in C^\infty_K([0,+\infty)\times\R)$,
%%
%\begin{eqnarray}
%%
%\int\int_{[0,+\infty)\times\R}h_c[\rho(t,x)]\partial_t\varphi(t,x)dxdt
%& + & 
%\int\int_{[0,+\infty)\times\R}g_c[\rho(t,x)]\partial_x\varphi(t,x)dxdt\nonumber\\
%%
%& + & \int_\R h_c[\rho_0(x)]dx\geq 0\label{entropy}
%%
%\end{eqnarray}
%%
%where $(h_c,g_c)$ is the Kru\v{z}kov's entropy-flux pair defined by
%%
%\be\label{kruzkov}
%%
%h_c(\rho)=(\rho-c)^+,\quad g_c(\rho)={\rm sgn}(\rho-c)[f(\rho)-f(c)]
%%
%\ee
%%
%It is equivalent to require \eqref{entropy} for $c$ in a dense subset (independent of $\varphi$) of $[0,1]$.\\ \\
%%
%%
%\textbf{Graphical (or Harris) construction.} The process with generator \eqref{generator} can be constructed as follows on
%the probability space $(\Omega_1,{\mathcal F}_1,\Prob_1)$, where 
%$\Omega_1$ is the set of space-time point measures $\omega_1$  of the form
%%
%$$
%\omega_1(dt,dx)=\sum_{n\in\N}\delta_{(t_n,x_n)}
%$$
%%
%equipped with the $\sigma$-field generated by the mappings $\omega\mapsto\omega((0,t]\times\{x\})$, where $t>0$ and $x\in\Z$, and the probability measure $\Prob_1$ that makes $\omega$ a random Poisson point process with intensity 
%$dt\otimes(\sum_{x\in\Z}\alpha(x)\delta_x)$.
%One can show that for a.e. realization of $\omega$, for any given initial state $\eta_0$, there exits a unique cadlag path $(\eta_t=\eta_t(\eta_0,\omega_1))_{t\geq 0}$ such that:\\ \\
%%
%(i) for every atom $(t,x)$ of $\omega_1$, $\eta_t=\eta_{t-}^{x,x+1}$ if $\eta_{t-}(x)=1$ and $\eta_{t-}(x+1)=0$, \\ \\
%%
%(ii) for every $x\in\Z$ and $0<s<t$, $\eta_.(x)$ is constant on $(s,t)$ if $\omega((s,t)\times\{x\})=0$.\\ \\
%%
%\margincom{Poisson periodique ?}
%The above construction yields a natural way of different TASEP's by using the same Poisson clocks.\\ \\
%%
%\textbf{Tightness and finite propagation.} The following tightness result can be found in \cite[Section 4]{rez} and 
%immediately adapted to our setting.
%%
%\begin{lemma}
%\label{lemma_tightness}
%%
%Let $\alpha\in {\bf A}$, and $(\eta^N_.)_{N\in\N^*}$ be a sequence of processes with generator \eqref{generator}. Let $\Pi^N_t:=\pi^N(\eta^N_{Nt})(dx)$.
%Then the sequence $(\Pi^N_.)_{N\in\N}$ of $\mathcal M(\R)$-valued processes is tight with respect to the Skorokhod topology on $\mathcal D([0,+\infty),\mathcal M(\R))$,
%and any subsequential limit in distribution is supported on $\mathcal C([0,+\infty),\mathcal M(\R))$.
%%
%\end{lemma}
%%
%\noindent The following classical finite propagation result states that discrepancies in two coupled TASEP's propagate at finite speed.
%%
%\begin{lemma}\label{lemma_finite_prop}
%%
%There exists a constant $V>0$ with the following property. Let $(\eta_.,\xi_.)$ be a coupled process, where $\eta_.$ has generator \eqref{generator}, 
%$\xi_.$ has generator \eqref{generator_torus_2}, and the two components are coupled as above. Let $a,b\in\R$ such that $0<b-a<1$. Then there exists a constant $C>0$
%such that
%%
%\be\label{finite_prop}
%%
%\Prob\left(
%%
%\eta_s(x)=\xi_s(x),\,\forall 
%{\color{black} s \in[0, N(b-a)/(2V)]}, \,\forall x\in[N(a+Vs),N(b-Vs)]\right)\leq C^{-1}e^{-CN}
%%
%%
%\ee
%%
%\end{lemma}
%%
%%
%%
%
%\medskip
%
%\textcolor{black}{Finally, we turn to }
%\begin{proof}[Proof of Theorem \ref{hdl_disorder}]\mbox{}\\ \\
%%
%{\em Step one.} In \cite{sep,bgrs4}, Theorem \ref{hdl_disorder} is proved for systems on $\Z$ with generator \eqref{generator} and deterministic Cauchy datum $\rho_0(.)$. Using these, we show how to deduce the case
%of random initial data.\\ \\
%%
%By Skorokhod's embedding theorem, there exists on some probability space  $(\Omega_0,\mathcal F_0,\Prob_0)$ a sequence of $\mathcal M(\R)$-valued random variables $(\tilde{\Pi}^N_0)_{N\in \N^*}$ 
%and a $\mathcal M(\R)$-valued random variable $\tilde{\Pi}_0$ such that:  
%%
%(i) $\tilde{\Pi}^N_0$ has the same distribution as $\Pi^N_0$; ii) $\tilde{\Pi}^N_0\to\tilde{\Pi}_0$ a.s. as $N\to\infty$.
%%
%Note that $\tilde{\Pi}_0$ is a.s. of the form $\tilde{\rho}_0(\omega_0)(x)dx$ for some random $\tilde{\rho}_0(\omega_0)\in L^\infty(\R;[0,1])$.
%Since $\pi^N$ is a one-to-one mapping from  ${\bf X}^N_{\rm per}$ to 
%$\mathcal M(\R)$, we have $\Pi^N=(\pi^N)(\tilde{\eta}^N_0)$, where $\tilde{\eta}^N_0$ is a ${\bf X}$-valued random variable with the same distribution as $\eta^N_0$. 
%%
%% For $\Prob_0$-a.e. $\omega\in\Omega$,
%% we have the determinstic vague  convergence 
%% %
%% \be\label{random_cauchy}\tilde{\eta}^N_0(\omega)\sim\rho_0(.)(\omega)\ee
%%
%We set
%%
%$$
%\tilde{\Pi}^N_t(\omega_0,\omega_1):=\pi^N_{Nt}\left(
%%
%\eta_t\left(\tilde{\eta}^N_0,\omega_1\right)
%%
%\right)
%$$  
%%
%Let $(\tilde{\Omega},\tilde{\mathcal F},\tilde{\Prob})$ denote the product of the probability spaces $(\Omega_0,\mathcal F_0,\Prob_0)$ and $(\Omega_1,\mathcal F_1,\Prob_1)$. Under $\tilde{\Prob}$, the process
%$\omega=(\omega_0,\omega_1)\mapsto\eta_.(\tilde{\eta}^N_0(\omega_0),\omega_1)$ has the same distribution as the original process with initial state $\eta^N_0$.
%Let $(S_t)_{t\geq 0}$ denote the evolution semigroup of \eqref{conservation_law} on 
%$L^\infty(\R;[0,1])$, i.e. for every 
%$u_0\in L^\infty(\R;[0,1])$, 
%$(t,x)\mapsto u_t(x):=S_t u_0(x)$ is the entropy solution to 
%\eqref{conservation_law} with Cauchy datum $u_0(.)$.
%%
%For fixed $\omega_0$, with randomness coming only from $\omega_1$, we can view $\tilde{\rho}_0(\omega_0)$ as a deterministic Cauchy datum for \eqref{conservation_law}.
%By  (ii) above and the results of \cite{sep} or \cite{bgrs4} for deterministic Cauchy datum, for $\Prob_0$-a.e. $\omega_0\in\Omega_0$, $\Pi^N_.(\omega_0,\omega_1)$ converges in $\Prob_1$-probability
%to $\Pi_.(\omega_0)$ defined by $\Pi_t(\omega_0)=S_t[\tilde{\rho_0}(\omega_0)](.)dx$. Thus $\tilde{\Pi}^N_.(\omega_0,\omega_1)$, which has the same distribution as $\Pi^N_.$, converges in $\Prob_0\otimes\Prob_1$-probability
%to $\Pi_.(\omega_0)$, which has the same distribution as $S_.[\rho_0(.)]dx$.
%%
%\\ \\
%%
%{\em Step two.}
%%
%We now consider the periodic case \eqref{generator_torus_2}.
%The result here cannot be obtained directly by the methods used in  \cite{sep} or \cite{bgrs4}. However it can be deduced a posteriori from
%the latter results on $\Z$ by coupling and localization arguments.
%%
%%
%%
%Let $\pi_t(dx)=\rho(t,x)dx$ be a limit in law of the rescaled empirical measure process $\Pi^N_.=\pi^N(\eta^N_{N.})$. We have to verify that, with probability $1$,  the random function $\rho(.,.)$ satisfies \eqref{entropy} for all $\varphi\in C^\infty_K([0,+\infty)\times\R)$. Let $D$ be a countable dense subset of $[0,+\infty)\times\R$. Using partitions of unity, it is enough to show the following: 
%for every $(t_0,x_0)\in D$, there exists a neighborhood $V(t_0,x_0)$ of $(t_0,x_0)$ in $[0,+\infty)\times\R$,
%such that \eqref{entropy} holds for every 
%$\varphi\in C^\infty_K\left(
%%
%V(t_0,x_0)
%%
%\right)$. Considering a countable dense subset $C$ of values of $c$, we only have to show that for every $(t_0,x_0,c)\in D\times C$, and every
%$\varphi\in C^\infty_K\left(
%%
%V(t_0,x_0)
%%
%\right)$, \eqref{entropy} holds with probability $1$.\\ \\
%%
%Let $t_1:=\max[0,t_0-\delta/(3V)]$. Choose $\delta<1/2$.
%We couple the process $\eta^N_.$ with a process $\xi^N_.$ defined at times $t\geq Nt_1$ with generator \eqref{generator}, and such that at time $Nt_1$ we have 
%$\eta^N_{Nt_1}(x)=\xi^N_{Nt_1}(x)$ for every $x\in\{0,\ldots,N-1\}$.  
%Lemma \ref{lemma_finite_prop} below, with $a=x_0-\delta$, $b=x_0+\delta$, 
%implies that for 
%$t\in(t_1,t_0+\delta/(3V))$, $\pi^N(\eta^N_{Nt})$ and $\pi^N(\xi^N_{Nt})$  coincide on  $(x_0-\delta/2,x_0+\delta/2)$. We then set 
%$V(t_0,x_0)=(t_1,t_0+\delta/(3V))\times(x_0-\delta/2,x_0+\delta/2)$. 
%%
%%By Corollary \ref{hdl_no_cauchy} applied to $\xi^N_.$,
%%
%Let us consider a subsequence along which $(\pi^N(\xi^N_{Nt_1})$ converges in distribution to some random measure $\tilde{\pi}_0=\tilde{\rho}_0(.)dx$ on $\R$.
%Any weak limit in law of  the joint process  $\left(\pi^N(\eta^N_{Nt}),\pi^N(\xi^N_{Nt})\right)_{t\in [t_1,t_0+\delta/(3V)]}$ 
%can be viewed as a random function $(\rho(t,x),\tilde{\rho}(t,x))$ such that 
%$\rho(t,x)=\tilde{\rho}(t,x)$ a.s. on $V(t_0,x_0)$.
%Along this subsequence we know by step one that the process $\big( \pi^N(\xi^N_{N(t_1+.)} ) \big) $ converges in distribution to the random path $\tilde{\pi}_.$ defined by
%$\tilde{\pi}_t(dx)=\tilde{\rho}(t,.)dx$, where $\tilde{\rho}(.,.)$ is the entropy solution to \eqref{conservation_law} with random Cauchy datum $\tilde{\rho}_0(.)$.
%Thus the random function $\tilde{\rho}(.,.)$, and therefore also $\rho(.,.)$, a.s. satisfies \eqref{entropy} on $V(t_0,x_0)$.
%If $t_1>0$, there is no initial term in \eqref{entropy}. 
%If $t_1=0$, then $\tilde{\rho}_0=\rho_0$ on $(x_0-\delta/2,x_0+\delta/2)$, and thus the initial term
%obtained with $\tilde{\rho}_0$ is the same as with $\rho_0$.
%%
%\end{proof}
%%
%%
%%
\section{Proof of Proposition \ref{prop_max}.}
\label{appendix_max}
%
%{\bf Ecrire la 2eme preuve basee sur def 1 et comparaison syst ouvert ds strech r}.
%
To show that the maximum value of the flux is at least $r/4$, we use Definition \textcolor{black}{(\ref{flux_current})} and couple the process $(\eta_t^\alpha)_{t\geq 0}$ with generator \eqref{generator}
with a homogeneous rate $r$ TASEP denoted by $(\eta_t^r)_{t\geq 0}$. A standard coupling argument 
 shows that $J_x^\alpha(t,\eta^\rho)\geq J_x^r(t,\eta^\rho)$, where $J_x^r$ denotes the current in the homogeneous rate $r$ TASEP.
It is known (see e.g. \cite{sep0}) that
%
$$
\lim_{t\to+\infty} \frac{1}{t} J_x^r(t,\eta^\rho)=r\rho(1-\rho)
$$
%
and  it is maximum for 
$\rho=1/2$.

\medskip


\margincom{\color{red} On a perdu l'autre definition du flux.\\ On la rajoute ?\textcolor{black}{bien oblige}}
We now prove that $f(\rho)\leq r/4$ for all $\rho\in[0,1]$.
\textcolor{black}
{
%
In \cite{bgrs4, bb} it is shown that there exists a closed subset $\mathcal R$  of $[0,1]$ containing $0$ and $1$,
and a family $(\nu^\alpha_\rho)_{\rho\in{\mathcal R}}$ of invariant measures for the disordered TASEP, such that, for every $\rho\in\mathcal R$,
%
\be\label{flux_line}
f(\rho)=\int j_x^\alpha(\eta)d\nu^{\alpha}_{\rho}(\eta),\quad x\in\Z,
%\{0,\ldots,N-1\},
\ee
%
%where $\nu^{\alpha}_{N,k_N}$ is the unique invariant measure for the process with $k_N$ particles on $\{0,\ldots,N-1\}$ with periodic boundary conditions, and $\lim_ {N\to+\infty}k_N/N=\rho$. Note that the integral in \eqref{flux_line} does not depend on $x$.
%
where $j_x^\alpha(\eta):=\alpha(x)\eta(x)[1-\eta(x+1)]$, and that $f$ is interpolated linearly outside $\mathcal R$. Note (this follows from stationarity) that the integral in \eqref{flux_line} does not depend on $x$.
%
}
It is thus enough to consider $\rho\in\mathcal R$.
%, where $\mathcal R$ is the density set in \eqref{flux_line}, since the %flux is interpolated linearly outside this set.
%We use definition \eqref{flux_current_3} of $f$. 
%
Since the random variables $\alpha(x)$ are i.i.d. and the infimum of their support is $r$,
for $\mathcal P$-a.e. environment $\alpha\in{\bf A}$, 
there exist sequences $(x_N)_{N \geq 1}$, $(y_N)_{N  \geq 1}$ and $(\varepsilon_N)_{N  \geq 1}$ such that \margincom{\textcolor{black}{on n'utilisait pas $x_N,y_N$ dans $\{0,\ldots,N-1\}$}}
%$x_N\in\{0,\ldots,N-1\}$, $y_N\in\{0,\ldots,N-1\}$,
$\lim_{N\to\infty}x_N=+\infty$, $\lim_{N\to\infty}[y_N-x_N]=+\infty$, $\lim_{N\to\infty}\varepsilon_N=0$, and
%
\be
\label{interval_r}
r\leq\min_{x=x_N,\ldots,y_N}\alpha(x)\leq\max_{x=x_N,\ldots,y_N}\alpha(x)\leq r+\varepsilon_N .
\ee
%
Set 
$$
a_N=\frac{2x_N+y_N}{3},\quad
b_N=\frac{x_N+2y_N}{3},
$$
which satisfy $x_N \leq a_N \leq b_N \leq y_N$ and $b_N-a_N\to+\infty$.
By \eqref{flux_line},
%
\begin{eqnarray}
%
f(\rho) =  \frac{1}{b_N-a_N+1}
\sum_{x=a_N}^{b_N}\int_{\bf X}j^\alpha_x(\eta)d\textcolor{black}{\nu^\alpha_\rho}(\eta)%\nonumber\\
\leq [r+\varepsilon_N] \int_{\bf X}\tilde{j}(\eta)d{\mu}_N (\eta),
\label{flux_inequality} 
\end{eqnarray}
%
where 
%
$
\tilde{j}(\eta)=\eta(0)[1-\eta(1)]
$, and 
%
\be
\label{averaged_measure}
{\mu}_N:= \frac{1}{b_N-a_N+1} \sum_{x=a_N}^{b_N}\tau_x\textcolor{black}{\nu^\alpha_\rho}.
\ee
%
The sequence $(\mu_N)_{N\in\N^*}$ of probability  on the compact space $\bf X$  is tight. Let $\mu^\star$ be one of its limit points. It follows from \eqref{averaged_measure} that $\mu^\star$
is shift invariant, i.e. $\tau_x\mu^\star = \mu^\star$ for all $x\in\Z$. We claim and prove below that $\mu^\star$ is an invariant measure for the homogeneous TASEP, that is the process with generator \eqref{generator} with $\alpha(x)\equiv 1$. By Liggett's characterization result \cite{lig} for shift-invariant stationary measures, $\mu^\star$ is then of the form
%
$$
\mu^\star  = \int_{[0,1]}\textcolor{black}{\nu_\rho}\gamma(d\rho).
$$
%
where $\gamma$ is a probability measure on $[0,1]$, and \textcolor{black}{$\nu_\rho$} is the product Bernoulli measure on $\bf X$ with parameter $\rho$. Thus 
%
$$
\int_{\bf X}\tilde{j}(\eta)d\mu^\star (\eta)=\int_{[0,1]}\rho(1-\rho)d\gamma(\rho)\leq\frac{1}{4}.
$$
%
Letting $N\to\infty$ in \eqref{flux_inequality} implies $f(\rho)\leq r/4$.\\ \\
%
%
We now prove that $\mu^\star$ is an invariant measure for the homogeneous TASEP. Let $g:{\bf X}\to\R$ be a local function that depends on $\eta$ only through sites $x\in\Z$ such that $|x|\leq\Delta$,
where $\Delta\in\N$. Take $N$ large enough so that
$\Delta<(y_N-x_N)/3$. Notice that  the generator $L^\alpha$ defined in \eqref{generator} satisfies the commutation relation
%
\be\label{commutation}
%
\tau_x  L^{\tau_x \alpha} f=L^\alpha (\tau_x f).
%
\ee
%
It follows that
%
$$
\int_{\bf X} \textcolor{black}{L^{\tau_x\alpha}} g\,d(\tau_x\textcolor{black}{\nu_\rho^\alpha})=\int_{\bf X} L^{\alpha}(\tau_x g)\,d\textcolor{black}{\nu_\rho^\alpha}=0.
$$
%
The last equality follows from invariance of $\nu_\rho^\alpha$. 
%
On the other hand, for $x\in[a_N,b_N]$ and $|y|\leq(y_N-x_N)/3$,  $\tau_x\alpha(y)\in[r,r+\varepsilon_N]$. Let $L$ denote the generator of the homogeneous TASEP on $\Z$,
that is the one obtained from \eqref{generator} when $\alpha(x)\equiv 1$. Since
%
$$
\Big| \textcolor{black}{L^{\tau_x\alpha}} g(\eta)-r Lg(\eta) \Big|
\leq  
2 ||g||_\infty \sum_{y=-\Delta-1}^{\Delta}|\tau_x\alpha(y)-r|,
$$
%
it follows that
%
$$
\lim_{N\to\infty} \max_{x=a_N,\ldots,b_N}\sup_{\eta\in{\bf X}}\left|
%
\textcolor{black}{L^{\tau_x\alpha}}g(\eta)-r Lg(\eta)
%
\right|=0.
$$
%
Hence
%
$$
\int_{\bf X}Lg(\eta) \, d \mu^\star (\eta)=\lim_{N\to\infty}\int_{\bf X}Lg(\eta)d\mu_N(\eta)
%
=\lim_{N\to+\infty}
%\int_{\bf X} \frac{1}{r} L^\alpha g(\eta)d\mu_N(\eta)
%
\textcolor{black}{\frac{1}{b_N-a_N}\sum_{x=a_N}^{b_N}
\int_{\bf X}\frac{1}{r}L^{\alpha}(\tau_x g)\,d\nu^\alpha_\rho}
%
=0
$$
%
holds for every local function $g$.
%
%
\section{Proof of Lemma \ref{lemma_reform}}
\label{appendix_lpp}
%
Before deriving  Lemma  \ref{lemma_reform}, we first explain a mapping between the restricted passage times in a box $B$ and the TASEP restricted to $B$ with reservoirs.



\subsection{last passage times in  a finite domain}
%
\margincom{
% 
\textcolor{black}{
%
Plusieurs modifications dans cette sous-section.
}
%
}
%
Let $B:=[x_1,x_2]\cap\Z$. The purpose of this subsection is to give an interpretation of the passage times \eqref{restricted_time} restricted to $B$ in terms of an open disordered TASEP
%We consider the open disordered TASEP 
on $B':=[x_1+1,x_2]\cap\Z$ with generator $L^\alpha_{B}$, see \eqref{generator_open} (recall from \eqref{def_bdiese} and \eqref{def_bprime} that
$(B')^\#=B$).
It is convenient to view the dynamics generated by \eqref{generator_open} as follows.
We add an infinite stack of particles (reservoir) at site $x_1$, and a site $x_2+1$ where the number of particles is not restricted.
Particles enter $B'$ from the stack at $x_1$, and when they leave, they stay at $x_2+1$ forever.
%Both $x_1-1$ and $x_2+1$ are infinite stacks which do not obey the exclusion rule.
%For the boundaries, we use the convention that particles to the left of $x_1$ are at $x_1-1$ and particles to the right of $x_2$ stay at $x_2+1$ once they reach this site. 
%
%
%
We are going to check that  $T^\alpha_B \big((x_1,0),(i,j) \big)$ has the same distribution as the time when particle $j$ reaches site $i+1$ in the process generated by $L^\alpha_B$,
if the initial state is given by
%
\be
\label{special_initial}
\sigma_0(j)=x_1\indicator{\{j\geq \textcolor{black}{0}\}}+(x_2+1)\indicator{\{j\leq-1\}}.
\ee
%
where $\sigma_0(j)$ denotes the initial position of  the particle with label $j$, and particles are numbered increasingly from right to left. 
%Whenever a particle enters $B'$ from $x_1-1$, the particle at $x_1-1$ with highest label is chosen. 
In fact,  we may define passage times associated with more general labeled initial configurations in $B'$. By this we mean that $\sigma_0$, instead of being defined by \eqref{special_initial},
can be any nonincreasing function $\sigma_0$ from $\Z$ to $[x_1,x_2+1]\cap\Z$.
% where $\sigma_0(j)$ denotes the initial position of the particle with label $j$.
%The position of the particle with label $j$ at time $t$, $j\in\Z$, is denoted by $\sigma_t(j)$. 
%
Let
%
%
\begin{equation}
\label{new_passage_bulk}
\tilde{B}  := \{ (i,j)\in B\times\Z:\,
i\geq\sigma_0(j) \}
% \label{new_passage_out}
\quad \text{and} \quad 
\bar{B} :=  \{ (i,j)\in B\times\Z:\,  i<\sigma_0(j)  \}.
\end{equation}
%
For $(i,j)\in B\times\Z$, let $T^\alpha_{B,\sigma_0} (i,j)$
%, where 
%
%$(i,j)\in\tilde{B}$,
%\times\N=:\tilde{B}$, 
%with $\tilde{B}$ defined in \eqref{new_passage_bulk} below,
%
denote the time at which particle $j$ reaches site $i+1$.
%
These passage times are determined by the boundary condition
%
\be\label{passage_boundary}
%%
T^\alpha_{B,\sigma_0} (i,j)=0\mbox{ for }(i,j)\in\bar{B}
%:=\Z^2\backslash\tilde{B},
%%
\ee
%%
%
together with the following recursions:
%
%Given $(i,j)\in B\times\N$, let
%%
%$$
%\mathcal N(i,j)=\{(i-1,j),(i+1,j-1)\}\cap(B\times\N)
%$$
%%
%Then
%
\begin{equation}
\label{recursion_passage}
T^\alpha_{B,\sigma_0} (i,j) =  \frac{Y_{i,j}}{\alpha(i)}+\max[T^\alpha_{B,\sigma_0} (i-1,j),
T^\alpha_{B,\sigma_0} (i+1,j-1)]
%\quad \mbox{ for }(i,j)\in\tilde{B},
\end{equation}
%
for $(i,j)\in\tilde{B}$ such that  $x_1<i<x_2$,
%
 \begin{equation}
 \label{recursion_passage_right}
 T^\alpha_{B,\sigma_0} (i,j) =  \frac{Y_{i,j}}{\alpha(i)}+T^\alpha_{B,\sigma_0} (i-1,j),
 %T^\alpha_{B,\sigma_0} (i+1,j-1)]
 %\quad \mbox{ for }(i,j)\in\tilde{B},
 \end{equation}
 %
 for  $(i,j)\in\tilde{B}$ such that $i=x_2$,
 %
 \begin{equation}
 \label{recursion_passage_left}
 T^\alpha_{B,\sigma_0} (i,j) =  \frac{Y_{i,j}}{\alpha(i)}+
 T^\alpha_{B,\sigma_0} (i+1,j-1),
 \end{equation}
 %
for  $(i,j)\in\tilde{B}$ such that $i=x_1$.
%
%%
%
%
In the special case \eqref{special_initial},
we have
%
\be
\label{old_bulk}
\tilde{B}  =  [x_1,x_2]\times\N
\quad \text{and} \quad 
\bar{B} = [x_1,x_2]\times(\Z\setminus\N).
\ee
%
By plugging \eqref{old_bulk} into \eqref{recursion_passage}, one recovers 
$$
T^\alpha_{B,\sigma_0}(i,j)=T^\alpha_B((x_1,0),(i,j)).
$$
where the r.h.s. was defined in \eqref{restricted_time}.
For notational simplicity, in the \textcolor{black}{sequel of this subsection}, we omit dependence on \textcolor{black}{$\alpha$}, $B$ and $\sigma_0$, and write $T(i,j)$ \textcolor{black}{instead of $T^\alpha_{B,\sigma_0}(i,j)$}.
%
%
The position of particle $j$ at time $t$, denoted by $\sigma_t(j)\in[x_1,x_2+1]$, is given by
%
\be
%
\label{position_time}
\sigma_t(j)  = \left\{
%
\ba{lll}
%
x_1 & \mbox{if} & T(x_1,j)>t\\
%
x_2+1 & \mbox{if} & T(x_2,j)\leq t\\
%
i\in[x_1+1,x_2]\cap\Z & \mbox{if} & T(i-1,j)\leq t<T(i,j)
%
\ea
%
\right.
%
\ee
%
\be
%
\label{time_position}
T(i,j)  =  \sup\{t\geq 0:\quad \sigma_t(j)\leq i\}.
%
\ee
%
%
%
%
The particle process $(\sigma_t)_{t\geq 0}$ is equivalent to the {\color{black} following growing cluster process:}
%$(\mathcal C_t)_{t\geq 0}$ defined by
%
$$
\mathcal C_t:=\{(i,j)\in [x_1,x_2]\times\Z:\,T(i,j)\leq t\}=\{(i,j)\in B\times\Z:\,i<\sigma_t(j)\}
$$
with initial state $\mathcal C_0=\bar{B}$. One can proceed as in \cite{sepcg} to show that both processes are Markovian and that the undistinguishable particle process $(\eta_t)_{t\geq 0}$ 
{\color{black} defined by}
%from $(\sigma_t)_{\textcolor{black}{t\geq 0}}$ via
%
\be\label{occupation}
\eta_t(x):=\sum_{j\in\Z}\indicator{\{\sigma_t(j)=x\}}
\ee
%
is Markov with generator $L^\alpha_{B}$.
%
%An alternative, slightly different interpretation of \eqref{restricted_time} in terms of the same TASEP is the following.
%%
%%
%Particles that leave $B'$ by jumping right from $x_2$ are sent immediately to $+\infty$,
%The latter is achieved by specifying that 
%%
%\be\label{outside_service_times}
%%
%Y_{i,j}=0\mbox{ if }i>x_2
%%
%\ee
%%
%while the random variables $Y_{i,j}$ for $i\leq x_2$ and $j\in\Z$ are still i.i.d. with distribution $\mathcal E(1)$.
%%
%%
%We label particles from right to left. 
%The position of label $j$ at time $t$ is denoted by $\sigma_t(j)$.
%%
%%
%We are going to check that  $T^\alpha_B \big((x_1,0),(i,j) \big)$ has the same distribution as the time when particle $j$ reaches site $i$ in the process generated by $L^\alpha_B$,
%if the initial state is given by
%%
%\be
%\label{special_initial}
%\sigma_0(j)=x_1\indicator{\{j\geq \textcolor{black}{0}\}}+(+\infty)\indicator{\{j\leq-1\}}.
%\ee
%%
%In fact,  passage times of paths restricted to $B$ with suitable boundary conditions can describe the evolution of the TASEP on $B'$ starting from any initial labeled particle configuration in $B'$,
%that is a nondecreasing function $\sigma_0$ from $\Z$ to $[x_1,+\infty]\cap\overline{\Z}$, where $\overline{\Z}:=\Z\cup\{\pm\infty\}$, and $\sigma_0(j)$ denotes the initial position of the particle with label $j$.
%%The position of the particle with label $j$ at time $t$, $j\in\Z$, is denoted by $\sigma_t(j)$. 
%Let $T^\alpha_{B,\sigma_0} (i,j)$, where 
%%
%$(i,j)\in\tilde{B}$,
%%\times\N=:\tilde{B}$, 
%with $\tilde{B}$ defined in \eqref{new_passage_bulk} below,
%%
%denote the time at which particle $j$ reaches site $i+1$. These times are now {\em always} determined by the recursions \eqref{recursion_passage}, but
%%
%%
%together with the boundary conditions
%%
%\be\label{passage_boundary}
%%
%T^\alpha_{B,\sigma_0} (i,j)=0\mbox{ for }(i,j)\in\bar{B}:=\Z^2\backslash\tilde{B},
%%
%\ee
%%
%where
%%
%\begin{eqnarray}
%\label{new_passage_bulk}
%%
%\tilde{B} &   := & \{
%%
%(i,j)\in \Z\times\Z:\,
%i\geq\sigma_0(j)
%%
%\}
%%
%\end{eqnarray}
%%
%For notational simplicity, in the following, we omit dependence on $B$ and $\sigma_0$, and write $T(i,j)$.
%%
%In the special case \eqref{special_initial},
%we have
%%
%\be
%%
%\label{old_bulk}
%%
%\tilde{B}  :=  ([x_1,+\infty)\cap\Z)\times\N.
%%
%\ee
%%
%By plugging \eqref{old_bulk} into \eqref{recursion_passage}, one recovers \eqref{restricted_time}. Indeed, it follows from \eqref{outside_service_times}
%that optimal paths never leave the box $B$, and moreover that $T(i,j)=T(x_2,j)$ for every $i>x_2$ and $j\in\Z$ (meaning that whenever a particle reaches site $x_2+1$ it instantly reaches all sites further right,
%i.e. is instantly sent to $+\infty$).
%%
%%
%The state of the system at a given time $t$ is equivalently described by the sequence $\sigma_t$ of particle positions, or the cluster
%%
%$$
%\mathcal C_t:=\{(i,j)\in\Z^2:\,T(i,j)\leq t\}.
%$$
%whose initial state is $\mathcal C_0=\overline{B}$.
%%
%There is a one-to-one correspondence between $\sigma_t$ and $\mathcal C_t$ given by
%%
%%
%\begin{eqnarray*}
%%
%\label{position_time}
%\sigma_t(j) & = & \max\{i\in[x_1-1,x_2+1]:
%\quad T(i,j)\leq t\},\\
%%
%\label{time_position}
%T(i,j) & = & \sup\{t\geq 0:\quad \sigma_t(j)\leq i\}.
%%
%\end{eqnarray*}
%%
%Thus the particle process $(\sigma_t)_{t\geq 0}$ is equivalent to the growing cluster process 
%$(\mathcal C_t)_{t\geq 0}$. One can proceed as in \cite{sepcg} to show that both processes are Markovian and that the undistinguishable particle process $(\eta_t)$ defined from $(\sigma_t)$ via
%%
%\be\label{occupation}
%\eta_t(x):=\sum_{j\in\Z}\indicator{\{\sigma_t(j)=x\}}
%\ee
%%
%is Markov with generator $L^\alpha_B$.
%
%
\subsection{Proof of Lemma \ref{lemma_reform}}
\ \\

\noindent
{\bf Step 1.} We prove that  definition \eqref{def_mintime} does not depend on $x_0$. Let $x_0,x'_0\in B$ with $x_0<x'_0$. Then \eqref{restricted_time} implies
%
\be\label{compare_times}
%
T^{\textcolor{black}{\alpha}}_B \big((x_0,0),(x_0,m+x'_0-x_0) \big)
\geq T^{\textcolor{black}{\alpha}}_B \big( (x'_0,0),(x'_0,m) \big) 
\geq T^{\textcolor{black}{\alpha}}_B \big((x_0,x'_0-x_0),(x_0,m) \big).
%
\ee
%
Since the sequence $(Y_{i,j}:\,i\in\Z,j\geq 0)$ is stationary with respect to shifts of $j$, the expectation of the last quantity is equal to that of
%
$T_B \big( (x_0,0),(x_0,m- x'_0 + x_0) \big)$. Thus taking expectations, dividing by $m$ and letting $m\to\infty$ yields the result.

\medskip

\noindent
{\bf Step 2.} {\it Proof of \eqref{max_current_lpp}.}
%
Given this statement, let us denote by $J^{\alpha,B}_x (t,\eta_0)$ the current up to time $t$ across site $x\in B'$, in the open system on $B'$, when starting from $\eta_0$. Assume $\eta_0$ is the occupation configuration associated with $\sigma_0$ via \eqref{special_initial}--\eqref{occupation}. Then it is clear that
%
%
$$
J^{\alpha,B}_x(t,\eta_0)=\min\{j\in\Z:\,T^{\alpha}_B(x,j)>t\}, 
$$
%
which implies the $\Prob$ a.s. limit
%
\begin{eqnarray*}
\lim_{t\to\infty}\frac{1}{t}J^{\alpha,B}_x(t,\eta_0) & = & \frac{1}{T_{\infty,B}}
 =  \lim_{t\to\infty}\Exp  \left( \frac{1}{t}J^{\alpha,B}_x(t,\eta_0)  \right)
 =   \lim_{t\to\infty} \Exp  \left( \frac{1}{t}\int_0^t j_x^{\alpha,B}(\eta_s)ds \right)\\
& = & \lim_{t\to\infty}\int j_x^{\alpha,B}(\eta)d\nu_t(\eta) 
=  \int j_x^{\alpha,B}(\eta)d\nu^\alpha_{B}(\eta),
\end{eqnarray*}
%
where
%
$$
j_x^{\alpha,B}(\eta)=\left\{
%
\ba{lll}
%
\alpha(x)\eta(x)[1-\eta(x+1)] & \mbox{if} & \textcolor{black}{x_1+1<x\leq x_2-1},\\
%
\alpha(x_2)\eta(x_2) & \mbox{if} & \textcolor{black}{x=x_2},\\
%
1-\eta(x_1+1) & \mbox{if} & \textcolor{black}{x=x_1+1},
%
\ea
%
\right.
$$
and 
$$
\nu_t:=\frac{1}{t}\int_0^t\delta_{\eta_0}e^{s L^\alpha_{B}}ds .
$$
%
The second equality follows from the fact that the family of random variables
 $(\frac{1}{t} J_x^{\alpha,B}(t,\eta_0))_{t\geq 0}$ is uniformly integrable, because 
 $\big( J_x^{\alpha,B}(t,\eta_0) \big)_{t\geq 0}$ is dominated in distribution by a Poisson random variable with parameter $t$.
The last equality follows from the fact that $\nu_t$ converges to the invariant measure $\nu^\alpha_{B}$ 
as $t$ tends to infinity.  \qed
%
%Indeed,
%the family $(\nu_t)_{t\geq 0}$ of probability measures on the compact space  $\{0,1\}^{B'}$  is tight.
%Any weak limit point of this family is an invariant measure for $L^\alpha_{B'}$. Finally, 
%the process is irreducible on its finite state space and has a unique invariant measure.\\ \\ 
%
\margincom{CH 31/10: enlev\'e l'appendice C preuve Lemme 2.1}
\section{Proof of Lemma \ref{cor_max}}\label{app:proof_cor_max}
%
\margincom{CH 31/10: ajout\'e preuve Lemme 7.3}
%
The proof of Lemma \ref{cor_max} relies on the following elementary estimates.
%
\begin{lemma}\label{lemma_normal}
\  \\
%
(i)
%
Let $Y$ be a random variable such that
%
$\Prob(Y\geq t)\leq Ce^{-t^2/V}$
%
for all $t\geq 0$, where $C\geq 1$ and $V>0$. Then, we have
%
$$
Y=\sqrt{V \log C}+\sqrt{V}X,
$$
%
where $\Prob(X\geq t)\leq e^{-t^2}$.

\medskip
%
%
\noindent
(ii) There exists a positive constant $A$ such that the following holds.
%
Let $(X_k)_{k=1,\ldots,n}$ be \textcolor{black}{independent} random variables such that
%
$\Prob(X_k\geq t)\leq e^{-t^2}$  for all $t\geq 0$,
%
and $(V_k)_{k=1,\ldots,n}$ be nonnegative numbers. 
%
Then 
%
$$
\sum_{k=1}^n \sqrt{V_k}X_k=\sqrt{\pi}\sum_{k=1}^n \sqrt{V_k}+\left(A\sum_{k=1}^n V_k\right)^{1/2}Z ,
$$
%
where $Z$ is a r.v. such that $\Prob(Z\geq t)\leq e^{-t^2}$  for all $t\geq 0$.
%and $\Exp Z\leq 0$.
%
%(iii) Let $\tilde{\Gamma}$ be a finite set, and $(Y_{\tilde{\gamma}})_{\tilde{\gamma}\in\tilde{\Gamma}}$ a family of random variables such that, for every $\tilde{\gamma}\in\tilde{\Gamma}$,
%%
%\be\label{hyp_normal}
%Y_{\tilde{\gamma}}\leq 
%%\Exp Y_\alpha
%m_{\tilde{\gamma}}+\sigma Z_{\tilde{\gamma}}
%\ee
%%
%where $m_{\tilde{\gamma}}\in\R$, $\sigma>0$, and $Z_{\tilde{\gamma}}$ is a random variable such that 
%%
%\be\label{normal_tail}\Prob(Z_{\tilde{\gamma}}\geq t)\leq e^{-t^2}\ee
%%
%for every $t\geq 0$. Then
%%
%\be\label{exp_max}
%%
%\Exp\max_{\tilde{\gamma}\in\tilde{\Gamma}}Y_{\tilde{\gamma}}\leq
%\max_{\tilde{\gamma}\in\tilde{\Gamma}}m_{\tilde{\gamma}}
%%\Exp Y_\alpha\leq 
%+\sigma\sqrt{\ln|\tilde{\Gamma}|}+\sigma\sqrt{\pi} 
%%
%\ee
%%
%where $|\tilde{\Gamma}|$ denotes the cardinality of $\tilde{\Gamma}$.
%
\end{lemma}
%
%
%
%\begin{remark}\label{remark_normal}
%%
%It is equivalent to assume that the random variable $Y_\alpha$ satisfies \eqref{hyp_normal}, or that there exists a constant $m_\alpha\in\R$, and a nonneagative random variable $Z_\alpha$ satisfying \eqref{normal_tail}, such that
%%
%\be\label{hyp_normal_bis}
%%
%Y_\alpha\leq m_\alpha+\sigma Z_\alpha
%%
%\ee
%%
%In particular, when assuming \eqref{hyp_normal_bis}, without loss of generality, one may suppose that $m_\alpha$
%%
%\end{remark}
%
\begin{proof}[Proof of Lemma \ref{lemma_normal}]
Assertion (i) follows from an immediate computation. To obtain (ii) we note that, for $\theta\geq 0$, 
\begin{eqnarray*}
\Exp \big( e^{\theta X_k} \big)  \leq  1+\int_0^{+\infty}\theta e^{\theta t}\Prob(X_k\geq t)dt
%\\& \leq & 
\leq 1+\theta e^{\theta^2/4}\int_{-\theta/2}^{+\infty}e^{-t^2}dt\leq1+\sqrt{\pi}\theta e^{\theta^2/4} \, .
\end{eqnarray*}
%
Setting $Y_k=X_k-\sqrt{\pi}$, we have, for $\theta\geq 0$,
%
$$
\Lambda(\theta) := \log \Exp \big( e^{\theta Y_k} \big) 
\leq  \log \left[
%
1+\sqrt{\pi}\theta e^{\theta^2/4}
%
\right]-\sqrt{\pi}\theta.
%
$$
%
Thus there exists $A>0$ such that $\Lambda(\theta)\leq A\theta^2/4$ for $\theta\geq 0$. Hence, by independence of the random variables $X_k$, we get
$$
\log \Exp \left[ \exp \left( \theta \left(
%
\sum_{k=1}^n \sqrt{V_k}X_k-\sqrt{\pi}\sum_{k=1}^n\sqrt{V_k} \right) 
%
\right) \right]
\leq \frac{A}{4}{\theta^2}\sum_{k=1}^n V_k.
$$
{\color{black} The estimate on the tail of $Z$ follows by an exponential Markov inequality.}
\end{proof}


%\medskip


\begin{proof}[Proof of Lemma \ref{cor_max}]
%
By (ii) of Lemma \ref{lemma_normal}, for every $a\in\mathcal A$, we have
%
\be\label{normal_1}
\sum_{i\in\mathcal I} {\mathcal Y}_{a,i}=
%m_a+(A V_a)^{1/2}Z_a
\sum_{i\in\mathcal I}\Exp \big( {\mathcal Y}_{a,i} \big)+\sqrt{\pi}\sum_{i\in\mathcal I}\sqrt{V_{a,i}}+\left(
%
A\sum_{i\in\mathcal I}V_{a,i}
%
\right)^{1/2}Z_a,
\ee
%
where
%%
%\begin{eqnarray}
%\label{def_ma}
%m_a & := & \Exp Y_{a,i}+\sqrt{\pi}\sum_{i\in\mathcal I}\sqrt{V_{a,i}}\\
%%
%V_a & := & \sum_{i\in\mathcal I}V_{a,i}\label{def_va}
%%
%\end{eqnarray}
%
%and 
$Z_a$ is a random variable satisfying $\Prob(Z_a\geq t)\leq e^{-t^2}$ for all $t\geq 0$. 
On the other hand, by Cauchy-Schwarz inequality,
%
\be
\label{cs_var}
%
\sum_{i\in\mathcal I}\sqrt{V_{a,i}}\leq\sqrt{|\mathcal I|}\left(\sum_{i\in\mathcal I}V_{a,i}\right)^{1/2}.
%
\ee
%
Thus, for every $a\in\mathcal A$,
%
\be\label{normal_2}
%
\sum_{i\in\mathcal I} {\mathcal Y}_{a,i}\leq m+(A\,V)^{1/2}Z^+_a,
%
\ee
%
where
%
\begin{eqnarray*}
%
V  := \max_{a\in\mathcal A}\sum_{i\in\mathcal I}V_{a,i}
\quad \text{and} \quad
m  :=  \sqrt{\pi}\sqrt{|\mathcal I|} \sqrt{V}
%\left(\max_{a\in\mathcal A}\sum_{i\in\mathcal I}V_{a,i} \right)^{1/2}
+\max_{a\in\mathcal A} \; \sum_{i\in\mathcal I} \Exp \big( Y_{a,i} \big) .
\end{eqnarray*}
%
Next, for any $t\geq 0$, we have
%
\begin{eqnarray*}
%
\Prob\left(
%
\max_{a\in\mathcal A}\sum_{i\in\mathcal I} {\mathcal Y}_{a,i}\geq m+t
%
\right) & \leq & \Prob\left(
%
\bigcup_{a\in\mathcal A}\left\{
%
\sum_{i\in\mathcal I} {\mathcal Y}_{a,i}\geq m+t
%
\right\}
%
\right)\\
%
& \leq & 
\sum_{a\in\mathcal A}
\Prob
\left(
(A\,V)^{1/2}Z^+_a\geq t
\right)
\leq |\mathcal A| \, e^{-\frac{t^2}{AV}} .
%
\end{eqnarray*}
%
It follows from (i) of Lemma \ref{lemma_normal} that
%
$$
\max_{a\in\mathcal A}\sum_{i\in\mathcal I} {\mathcal Y}_{a,i}= m+\sqrt{\log |\mathcal A|}(A\,V)^{1/2} +(A\,V)^{1/2}Z,
$$
%
where $Z$ is a random variable satisfying  $\Prob(Z\geq t)\leq e^{-t^2}$ for all $t\geq 0$. The result then follows from
%
$$
\Exp(Z)\leq\Exp(Z^+)\leq\int_0^{+\infty}\Prob(Z^+\geq t)dt\leq \int_0^{+\infty}e^{-t^2}dt=\sqrt{\pi}.
$$
%
\end{proof}
%
%
\end{appendix}
%
\begin{thebibliography}{9}
%
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Ahlberg, D.,  Damron, M., Sidoravicius, V.:
Inhomogeneous first-passage percolation. preprint 
arXiv:1311.4058 (2013)
%
\bibitem{AW}
Aizenman, M., Wehr, J.:  Rounding effects of quenched randomness on first-order phase transitions. 
{  Comm. Math. Phys.} {\bf 130}, Number 3, 489--528  (1990)
%
\bibitem{afgl}
Andjel, E., Ferrari, P.A., Guiol, H., Landim, C.:
Convergence to the maximal invariant measure for a zero-range process with random
rates. \text{Stoch. Proc. Appl.} \textbf{90},  67--81 (2000)
%
%
%\bibitem{acs} Armstrong, S.N., Cardaliaguet, P., Souganidis, P.E.: Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations.
%J. Amer. Math. Soc. {\bf 27}, 479-540 (2014) 
%
%\bibitem{bgrs2} Bahadoran, C., Guiol, H., Ravishankar, K., Saada, E. Euler hydrodynamics of one-dimensional attractive particle
%systems. Ann. Probab. {\bf 34},1339-1369 (2006)
%
\bibitem{bgrs4} Bahadoran, C., Guiol, H., Ravishankar, K., Saada, E.: Euler hydrodynamics for attractive particle
systems in random environment.  Ann. Institut Henri Poincar\'e {\bf 50}, 403-424 (2014)
%
\bibitem{bmrs} Bahadoran, C., Mountford, T., Ravishankar, K., Saada, E.: Supercriticality conditions  for asymmetric zero-range process with sitewise disorder. 
Brazilian Journal of Probability and Statistics. \textbf{29}, Number 2, 282--312 (2015) 
%
%\bibitem{bmrs2} Bahadoran, C., Mountford, T., Ravishankar, K., Saada, E.: Supercritical behaviour of asymmetric zero-range process with sitewise disorder.
% To appear in Ann. Instit. Poincaré.
%
%
\bibitem{bb} Bahadoran, C., Bodineau, T.: Properties and conjectures for the flux 
of TASEP with site disorder. Brazilian Journal of Probability and Statistics, \textbf{29}, Number 2,
282-312  (2015) 
%
%
\bibitem{bss} Basu, R.,  Sidoravicius, V.,  Sly A.: Last Passage Percolation with a Defect Line and the Solution of the Slow Bond Problem.
Math.PR 1408.3464.
%
\bibitem{bfl} Benjamini, I., Ferrari, P.A., Landim, C.: Asymmetric processes with random rates. Stochastic Processes and their Applications {\bf 61}, 181-204 (1996)
%
\bibitem{bt} Bodineau, T., Teixeira, A: Interface Motion in Random Media. 
Comm. Math. Phys. {\bf 334}  Number 2, 843-865 (2015)

%\bibitem{bricmontkup}
%Bricmont, J., Kupiainen, A.: Random walks in asymmetric random environments. Communications in mathematical physics, {\bf 142} Number 2, 345-420  (1991)
%
%
\bibitem{CL}
Chayes, L., Liggett, T.:
{One Dimensional Nearest Neighbor Exclusion Processes in Inhomogeneous and Random Environments}. Jour. Stat. Phys. {\bf 129}, Number 2, 193-203  (2007)
%
\bibitem{css} Chowdhury, D., Santen, L., Schadschneider, A.: Statistical physics of vehicular traffic and some related models. Physics Reports {\bf 329}, 199--329 (2000)
%
\bibitem{ED}
Derrida, B., Enaud, C.: Sample-dependent phase transitions in disordered exclusion models. 
Europhys. Lett. {\bf 66}, 83-89 (2004)
%
\bibitem{der} Derrida, B.,  Hakim, V., Evans, M.R., Pasquier, V.: Exact solution of a 1d asymmetric exclusion model using a matrix formulation J. Phys. A {\bf 26}, 1493-1517 (1992)
%
\bibitem{ev} Evans, M.R.: Bose-Einstein condensation in disordered exclusion models and relation to traffic flow. Europhysics Letters, {\bf 36}, Number 1, 13 (1996)
%
\bibitem{frt} Faggionato, A., Roberto, C. , Toninelli, C.: Universality for one-dimensional hierarchical coalescence processes with double and triple merges.
Ann. Appl. Probab. {\bf 24}, 476-525 (2014)
%
%\bibitem{fp} Ferrari, P., Pimentel, L.: Competition interfaces and second class particles. Ann. Probab. {\bf 33}, Number 4 1235-1254 (2005)
%
%
%
%\bibitem{lig0} Liggett, T.M.: Ergodic theorems for the simple exclusion process. Ann. Probab. {\bf 5}, 795-801 (1975)
%%
\bibitem{lig} Liggett, T.M.: Coupling the simple exclusion process. Ann. Probab. {\bf 3}, 339-356 (1976)
%
%
\bibitem{fs}  Ferrari, P. A., Sisko, V.: Escape of mass in zero-range processes
with random rates. \textit{Asymptotics: particles, processes and
inverse problems}, 108--120, IMS Lecture Notes Monogr. Ser.,
\textbf{55}, \textit{Inst. Math. Statist., Beachwood, OH}, 2007
%
%\bibitem{fk} Ferrari, P., Krug, J.: Phase transitions in driven diffusive systems with
%random rates. Journal of Physics A  {\bf 29}, Number 18,  L465 (1996)
%
\bibitem{GKS}
Georgiou, N., Kumar, R.,  Sepp\"al\"ainen, T.:
{TASEP with Discontinuous Jump Rates}.
ALEA Lat. Am. J. Probab. Math. Stat., {\bf 7}, 293-318  (2010)
%
\bibitem{GKS}
Grigorescu, I., Kang, M., Sepp\"al\"ainen, T. :
Behavior dominated by slow particles in a disordered asymmetric exclusion process. Ann. Appl. Probab. {\bf 14}, no. 3, 1577--1602 (2004)
%
%\bibitem{gl} Godr\`eche, C., Luck, J.M.: Condensation in the inhomogeneous zero-range
%process: an interplay between interaction and diffusion disorder. J. Stat. Mech. (2012) 
%
\bibitem{HS}
Harris, R.,  Stinchcombe, R.:
{Disordered asymmetric simple exclusion process: Mean-field treatment},  Phys. Rev. E {\bf 70}, 016108 (2004)
%
\bibitem{jl} Janowski, A., Lebowitz, J.L.: Finite-size effects and shock fluctuations in the asymmetric simple-exclusion process. 
Phys. Rev. A {\bf 45} (1992)
%
%
\bibitem{kru} Krug, J.: Phase separation in disordered exclusion models.
Brazilian Jour.  Phys.  {\bf 30}, 97-104 (2000)
%
\bibitem{khg} Khoromskaia, D., Harris, R.J., Grosskinsky, S.: Dynamics of non-Markovian exclusion processes. Preprint Arxiv: 1408.2139 (2014)
%
\bibitem{khh} Krug, J., Halpin-Healy, T.: 
{Directed polymers in the presence of columnar disorder}.
Jour. de Physique I {\bf 3}, Number 11, 2179-2198 (1993)
%
\bibitem{ks} Krug, J., Sepp\"al\"alainen, T.: Hydrodynamics and platoon formation for a totally asymmetric exclusion process with particlewise disorder.
J. Stat. Phys. {\bf 95} (1999)
%
%\bibitem{KSV}
%Kesten, H., Sidoravicius, V., Vares,  M.:
%{Oriented percolation in a random environment}, preprint (2012)
%
%\bibitem{lan} Landim, C.: Metastability for a non-reversible dynamics: the evolution of the condensate in totally asymmetric zero range processes.
%Commun. Math. Phys. {\bf 330} (2014) 
%
\bibitem{mar} Martin, J.: 
Limiting shape for directed percolation models.
Annals of Probability {\bf 32}, 2908-2937 (2004) 
%last passage percolation with general weight distribution.
%
\bibitem{rez} Rezakhanlou, F.: Hydrodynamic limit for attractive particle systems on $\Z^d$. Commun. Math. Phys. {\bf 140}, 417-448 (1991)
%
\bibitem{rez2} Rezakhanlou, F.: Continuum limit for some growth models II. Ann. Probab. {\bf 29}, 1329-1372 (2001)
%
\bibitem{sch1} Sch\"utz, G.:
{Generalized Bethe ansatz solution of a one-dimensional asymmetric exclusion process on a ring with blockage}.
J. Stat. Phys. \textbf{71}(3)-(4), 471--505 (1993).
%
%\bibitem{rez3} Rezakhanlou, F.: 
%{\it Continuum limit for some growth models II},  
%Annals of probability, 1329-1372 (2001).
\bibitem{sch2} Sch\"utz, G.: Conditioned stochastic particle systems and integrable quantum spin systems.   Preprint arXiv:1410.0184  (2014).
%
\bibitem{schd} Sch\"utz, G., Domany, E.: 
{Phase transitions in an exactly soluble one-dimensional asymmetric exclusion model}. J. Stat. Phys. {\bf 72}, 277-296 (1993).
%
%\bibitem{sepbook} Sepp\"al\"ainen, T.: Translation invariant exclusion processes. Unpublished manuscript.
%
\bibitem{sep0} Sepp\"al\"ainen, T.: Coupling the totally asymmetric exclusion process with a moving interface. Markov Process. Rel. Fields {\bf 4}, 592--628 (1998)
%
\bibitem{sep} Sepp\"al\"ainen, T.: Existence of Hydrodynamics for the Totally Asymmetric Simple K-Exclusion Process. 
Ann. Probab. {\bf 27}, Number 1 361-415 (1999)
%
\bibitem{sepslow} Sepp\"al\"ainen, T.: Hydrodynamic profiles for the totally asymmetric exclusion process with a slow bond. J. Stat. Phys. {\bf 102}, 69--96 (2001)
%
\bibitem{sepcg} Sepp\"al\"ainen, T.: Lecture notes on Corner growth model (2009)
%
\bibitem{sly} Sly, A. : Note on the flux for TASEP with general disorder, preprint arXiv:1609.06589 (2016)
%
\bibitem{sz} Szavits-Nossan, J. :
Disordered exclusion process revisited: some exact results in the low-current regime.
J. Phys. A: Math. Theor. \textbf{46} 315001 (2013)
%
\bibitem{Sznitman}
Sznitman, A. S.: Brownian motion, obstacles and random media. Springer  (1998)
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%\bibitem{Sznitman2}
%Sznitman, A. S.: Random motions in random media. in Mathematical Statistical Physics, A. Bovier, F. Dunlop, F. den Hollander, A. van Enter and J. Dalibard, eds., Les Houches, Session LXXXIII, 2005, Elsevier, 219-242 (2006)
%%
\bibitem{tal} Talagrand, M.: Concentration of measure and isoperimetric inequalities in product spaces. Publications IHES {\bf 81}, 73-205 (1995)
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\bibitem{tb} Tripathy, G., Barma, M.:
Driven lattice gases with quenched disorder: Exact results and different macroscopic regimes.
Phys. Rev. E {\bf 58}, 1911 (1998)
%

\bibitem{vares}
Sidoravicius, V., Vares, M.E.:
Interacting Particle Systems: Renormalization and Multi-scale Analysis,
%  author={ and Centro de Estudios Avanzados - Instituto Venezolano de Investigaciones Cientificas},
%  url={http://books.google.com.br/books?id=0lFpGwAACAAJ},
Asociacion Matematica Venezolana / Instituto Venezolano de Investigaciones Cientificas, (2005)


\end{thebibliography}
%
\end{document}
%
%=
1602.06718.extracted_bib
{ADV Ahlberg, D., Damron, M., Sidoravicius, V.: Inhomogeneous first-passage percolation. preprint arXiv:1311.4058 (2013) %
1602.06718.extracted_bib
{AW Aizenman, M., Wehr, J.: Rounding effects of quenched randomness on first-order phase transitions. { Comm. Math. Phys. {130, Number 3, 489--528 (1990) %
1602.06718.extracted_bib
{afgl Andjel, E., Ferrari, P.A., Guiol, H., Landim, C.: Convergence to the maximal invariant measure for a zero-range process with random rates. \text{Stoch. Proc. Appl. 90, 67--81 (2000) % % %
1602.06718.extracted_bib
{acs Armstrong, S.N., Cardaliaguet, P., Souganidis, P.E.: Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations. %J. Amer. Math. Soc. {27, 479-540 (2014) % %
1602.06718.extracted_bib
{bgrs2 Bahadoran, C., Guiol, H., Ravishankar, K., Saada, E. Euler hydrodynamics of one-dimensional attractive particle %systems. Ann. Probab. {34,1339-1369 (2006) %
1602.06718.extracted_bib
{bgrs4 Bahadoran, C., Guiol, H., Ravishankar, K., Saada, E.: Euler hydrodynamics for attractive particle systems in random environment. Ann. Institut Henri Poincar\'e {50, 403-424 (2014) %
1602.06718.extracted_bib
{bmrs Bahadoran, C., Mountford, T., Ravishankar, K., Saada, E.: Supercriticality conditions for asymmetric zero-range process with sitewise disorder. Brazilian Journal of Probability and Statistics. 29, Number 2, 282--312 (2015) % %
1602.06718.extracted_bib
{bmrs2 Bahadoran, C., Mountford, T., Ravishankar, K., Saada, E.: Supercritical behaviour of asymmetric zero-range process with sitewise disorder. % To appear in Ann. Instit. Poincaré. % %
1602.06718.extracted_bib
{bb Bahadoran, C., Bodineau, T.: Properties and conjectures for the flux of TASEP with site disorder. Brazilian Journal of Probability and Statistics, 29, Number 2, 282-312 (2015) % %
1602.06718.extracted_bib
{bss Basu, R., Sidoravicius, V., Sly A.: Last Passage Percolation with a Defect Line and the Solution of the Slow Bond Problem. Math.PR 1408.3464. %
1602.06718.extracted_bib
{bfl Benjamini, I., Ferrari, P.A., Landim, C.: Asymmetric processes with random rates. Stochastic Processes and their Applications {61, 181-204 (1996) %
1602.06718.extracted_bib
{bt Bodineau, T., Teixeira, A: Interface Motion in Random Media. Comm. Math. Phys. {334 Number 2, 843-865 (2015) %
1602.06718.extracted_bib
{bricmontkup %Bricmont, J., Kupiainen, A.: Random walks in asymmetric random environments. Communications in mathematical physics, {142 Number 2, 345-420 (1991) % %
1602.06718.extracted_bib
{CL Chayes, L., Liggett, T.: {One Dimensional Nearest Neighbor Exclusion Processes in Inhomogeneous and Random Environments. Jour. Stat. Phys. {129, Number 2, 193-203 (2007) %
1602.06718.extracted_bib
{css Chowdhury, D., Santen, L., Schadschneider, A.: Statistical physics of vehicular traffic and some related models. Physics Reports {329, 199--329 (2000) %
1602.06718.extracted_bib
{ED Derrida, B., Enaud, C.: Sample-dependent phase transitions in disordered exclusion models. Europhys. Lett. {66, 83-89 (2004) %
1602.06718.extracted_bib
{der Derrida, B., Hakim, V., Evans, M.R., Pasquier, V.: Exact solution of a 1d asymmetric exclusion model using a matrix formulation J. Phys. A {26, 1493-1517 (1992) %
1602.06718.extracted_bib
{ev Evans, M.R.: Bose-Einstein condensation in disordered exclusion models and relation to traffic flow. Europhysics Letters, {36, Number 1, 13 (1996) %
1602.06718.extracted_bib
{frt Faggionato, A., Roberto, C. , Toninelli, C.: Universality for one-dimensional hierarchical coalescence processes with double and triple merges. Ann. Appl. Probab. {24, 476-525 (2014) % %
1602.06718.extracted_bib
{fp Ferrari, P., Pimentel, L.: Competition interfaces and second class particles. Ann. Probab. {33, Number 4 1235-1254 (2005) % % % %
1602.06718.extracted_bib
{lig0 Liggett, T.M.: Ergodic theorems for the simple exclusion process. Ann. Probab. {5, 795-801 (1975) %%
1602.06718.extracted_bib
{lig Liggett, T.M.: Coupling the simple exclusion process. Ann. Probab. {3, 339-356 (1976) % %
1602.06718.extracted_bib
{fs Ferrari, P. A., Sisko, V.: Escape of mass in zero-range processes with random rates. Asymptotics: particles, processes and inverse problems, 108--120, IMS Lecture Notes Monogr. Ser., 55, Inst. Math. Statist., Beachwood, OH, 2007 % %
1602.06718.extracted_bib
{fk Ferrari, P., Krug, J.: Phase transitions in driven diffusive systems with %random rates. Journal of Physics A {29, Number 18, L465 (1996) %
1602.06718.extracted_bib
{GKS Georgiou, N., Kumar, R., Sepp\"al\"ainen, T.: {TASEP with Discontinuous Jump Rates. ALEA Lat. Am. J. Probab. Math. Stat., {7, 293-318 (2010) %
1602.06718.extracted_bib
{GKS Grigorescu, I., Kang, M., Sepp\"al\"ainen, T. : Behavior dominated by slow particles in a disordered asymmetric exclusion process. Ann. Appl. Probab. {14, no. 3, 1577--1602 (2004) % %
1602.06718.extracted_bib
{gl Godr\`eche, C., Luck, J.M.: Condensation in the inhomogeneous zero-range %process: an interplay between interaction and diffusion disorder. J. Stat. Mech. (2012) %
1602.06718.extracted_bib
{HS Harris, R., Stinchcombe, R.: {Disordered asymmetric simple exclusion process: Mean-field treatment, Phys. Rev. E {70, 016108 (2004) %
1602.06718.extracted_bib
{jl Janowski, A., Lebowitz, J.L.: Finite-size effects and shock fluctuations in the asymmetric simple-exclusion process. Phys. Rev. A {45 (1992) % %
1602.06718.extracted_bib
{kru Krug, J.: Phase separation in disordered exclusion models. Brazilian Jour. Phys. {30, 97-104 (2000) %
1602.06718.extracted_bib
{khg Khoromskaia, D., Harris, R.J., Grosskinsky, S.: Dynamics of non-Markovian exclusion processes. Preprint Arxiv: 1408.2139 (2014) %
1602.06718.extracted_bib
{khh Krug, J., Halpin-Healy, T.: {Directed polymers in the presence of columnar disorder. Jour. de Physique I {3, Number 11, 2179-2198 (1993) %
1602.06718.extracted_bib
{ks Krug, J., Sepp\"al\"alainen, T.: Hydrodynamics and platoon formation for a totally asymmetric exclusion process with particlewise disorder. J. Stat. Phys. {95 (1999) % %
1602.06718.extracted_bib
{KSV %Kesten, H., Sidoravicius, V., Vares, M.: %{Oriented percolation in a random environment, preprint (2012) % %
1602.06718.extracted_bib
{lan Landim, C.: Metastability for a non-reversible dynamics: the evolution of the condensate in totally asymmetric zero range processes. %Commun. Math. Phys. {330 (2014) %
1602.06718.extracted_bib
{mar Martin, J.: Limiting shape for directed percolation models. Annals of Probability {32, 2908-2937 (2004) %last passage percolation with general weight distribution. %
1602.06718.extracted_bib
{rez Rezakhanlou, F.: Hydrodynamic limit for attractive particle systems on $\Z^d$. Commun. Math. Phys. {140, 417-448 (1991) %
1602.06718.extracted_bib
{rez2 Rezakhanlou, F.: Continuum limit for some growth models II. Ann. Probab. {29, 1329-1372 (2001) %
1602.06718.extracted_bib
{sch1 Sch\"utz, G.: {Generalized Bethe ansatz solution of a one-dimensional asymmetric exclusion process on a ring with blockage. J. Stat. Phys. 71(3)-(4), 471--505 (1993). % %
1602.06718.extracted_bib
{rez3 Rezakhanlou, F.: %{Continuum limit for some growth models II, %Annals of probability, 1329-1372 (2001).
1602.06718.extracted_bib
{sch2 Sch\"utz, G.: Conditioned stochastic particle systems and integrable quantum spin systems. Preprint arXiv:1410.0184 (2014). %
1602.06718.extracted_bib
{schd Sch\"utz, G., Domany, E.: {Phase transitions in an exactly soluble one-dimensional asymmetric exclusion model. J. Stat. Phys. {72, 277-296 (1993). % %
1602.06718.extracted_bib
{sepbook Sepp\"al\"ainen, T.: Translation invariant exclusion processes. Unpublished manuscript. %
1602.06718.extracted_bib
{sep0 Sepp\"al\"ainen, T.: Coupling the totally asymmetric exclusion process with a moving interface. Markov Process. Rel. Fields {4, 592--628 (1998) %
1602.06718.extracted_bib
{sep Sepp\"al\"ainen, T.: Existence of Hydrodynamics for the Totally Asymmetric Simple K-Exclusion Process. Ann. Probab. {27, Number 1 361-415 (1999) %
1602.06718.extracted_bib
{sepslow Sepp\"al\"ainen, T.: Hydrodynamic profiles for the totally asymmetric exclusion process with a slow bond. J. Stat. Phys. {102, 69--96 (2001) %
1602.06718.extracted_bib
{sepcg Sepp\"al\"ainen, T.: Lecture notes on Corner growth model (2009) %
1602.06718.extracted_bib
{sly Sly, A. : Note on the flux for TASEP with general disorder, preprint arXiv:1609.06589 (2016) %
1602.06718.extracted_bib
{sz Szavits-Nossan, J. : Disordered exclusion process revisited: some exact results in the low-current regime. J. Phys. A: Math. Theor. 46 315001 (2013) %
1602.06718.extracted_bib
{Sznitman Sznitman, A. S.: Brownian motion, obstacles and random media. Springer (1998) % %
1602.06718.extracted_bib
{Sznitman2 %Sznitman, A. S.: Random motions in random media. in Mathematical Statistical Physics, A. Bovier, F. Dunlop, F. den Hollander, A. van Enter and J. Dalibard, eds., Les Houches, Session LXXXIII, 2005, Elsevier, 219-242 (2006) %%
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{tal Talagrand, M.: Concentration of measure and isoperimetric inequalities in product spaces. Publications IHES {81, 73-205 (1995) %
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{tb Tripathy, G., Barma, M.: Driven lattice gases with quenched disorder: Exact results and different macroscopic regimes. Phys. Rev. E {58, 1911 (1998) %
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{vares Sidoravicius, V., Vares, M.E.: Interacting Particle Systems: Renormalization and Multi-scale Analysis, % author={ and Centro de Estudios Avanzados - Instituto Venezolano de Investigaciones Cientificas, % url={http://books.google.com.br/books?id=0lFpGwAACAAJ, Asociacion Matematica Venezolana / Instituto Venezolano de Investigaciones Cientificas, (2005)