Incomplete determinantal processes: from random matrix to Poisson statistics
Gaultier Lambert\thanks{KTH Royal Institute of Technology
Department of Mathematics. Supported by the grant KAW 2010.0063 from the Knut and Alice Wallenberg Foundation.
\noindent \normalsize We study linear statistics of a class of determinantal processes which interpolate between Poisson and GUE/Ginibre statistics in dimension 1 or~2. These processes are obtained by performing an independent Bernoulli percolation on the particle configuration of a Coulomb gas with logarithmic interaction confined in a general potential. We show that, depending on the expected number of deleted particles, there is a universal transition for mesoscopic statistics. Namely, at small scales, the point process still behaves like a Coulomb gas, while, at large scales, it needs to be renormalized because the variance of any linear statistic diverges. The crossover is explicitly characterized as the superposition of a $H^{1/2$- or $H^1$- correlated Gaussian noise and an independent Poisson process. The proof consists in computing the limits of the cumulants of linear statistics using the asymptotics of the correlation kernel of the process.
1612.00806.tex
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%\title{A transition from random matrix to Poisson statistics for determinantal processes}
\title{Incomplete determinantal processes: from random matrix to Poisson statistics}
\date{}
\author{Gaultier Lambert\thanks{KTH Royal Institute of Technology, Department of Mathematics. Supported by the grant KAW 2010.0063 from the Knut and Alice Wallenberg Foundation.}\ \thanks{Department of Mathematics, University of Zurich, gaultier.lambert@math.uzh.ch}} 


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\begin{document}

\maketitle

\begin{abstract}
\noindent \normalsize
We study linear statistics of a class of determinantal processes which interpolate between Poisson and GUE/Ginibre statistics in dimension 1 or~2. These processes are obtained by performing an independent Bernoulli percolation on the particle configuration of a Coulomb gas with logarithmic interaction confined in a general potential. 
We show that, depending on the expected number of deleted particles, there is a universal  transition for mesoscopic statistics. Namely, at small scales, the point process still behaves like a Coulomb gas, while, at large scales, it needs to be renormalized because the variance of any linear statistic diverges. The crossover is explicitly characterized as the superposition of a $H^{1/2}$- or $H^1$- correlated  Gaussian noise and an independent Poisson process. 
The proof consists in computing the limits of the cumulants of linear statistics using the asymptotics of the correlation kernel of the process.
\end{abstract}
 
  \section{introduction}
  
In this notes, we consider a log-gas,  also called $\beta$-ensemble or one component plasma, in dimension $1$ or $2$ at inverse temperature $\beta=2$. Let $\X = \R$ or $\C$ equipped with the Lebesgue measure  $d\mu = dx$ or the  area measure 
$d\mu= d\A = \frac{r dr d\theta}{\pi}$ respectively, and let $V: \X \to\R$ be a continuous function such that there exists $\nu >0$ so that 
\begin{equation} \label{potential}
 V(z) \ge (1+\nu) \log|z|   
\hspace{.5cm}\text{as}\hspace{.4cm} |z| \to\infty .
\end{equation}
We consider the probability measure on $\X^N$ with  density $G_N(x)=e^{-\beta \mathscr{H}^N_V(x)}/Z^N_V$ where the Hamiltonian is
 \begin{equation} \label{log_gas}
\mathscr{H}^N_V(x)=  \hspace{-.15cm} \sum_{1\le i<j \le N}   \hspace{-.15cm}\log| x_i -x_j|^{-1}  + N \sum_{j=1}^N V(x_j) .
 \end{equation}   
 The condition $\beta=2$ implies that if the configuration $(\lambda_1,\dots, \lambda_N)$ is distributed according to $G_N$, then the point process $\Xi := \sum_{k=1}^N \delta_{\lambda_k}$ is  determinantal  with a correlation kernel:
  \begin{equation} \label{kernel}
K^N_V(z,w) = \sum_{k=0}^{N-1} \varphi_k(z) \varphi_k(w)   
\end{equation} 
with respect to $\mu$. Moreover, for all $k\ge 0$, 
\begin{equation} \label{OP}
 \varphi_k(x) = P_k(x) e^{-N V(x)} ,
\end{equation}
 where  $\{P_k \}_{k=0}^\infty$ is the sequence of orthonormal polynomials with respect to the weight $e^{- 2N V(x)}$ on $L^2(\mu)$. 
It turns out that when $\beta=2$,  the Hamiltonian \eqref{log_gas} also describes the eigenvalues of the ensembles of Hermitian (or complex normal) matrices with weight $e^{-2NV(M)}$ on $\X=\R$ (or $\X=\C$).   
 In particular, when $V(z)=|z|^2$, these correspond to the well-know Gaussian Unitary (GUE) or Ginibre ensembles respectively. \\  
%For large classes of test functions, the asymptotic properties of linear statistics of these ensembles have been investigated and laws of large numbers (LLN) and central limit theorems (CLT) have been proved in various regime, see \cite{Johansson98, AHM11, AHM15,  BD14, BD16, Lambert_b, BL16} to cite of few of the main contributions or the books \cite{Forrester10, PS11} for a review of the theory. \\
 
  
 In the following, we  consider the so-called {\it incomplete} or {\it thinned point  process} which arises after performing a Bernoulli percolation on the configuration of the process $\Xi$. This process, denoted by $\hat\Xi$, is obtained by deleting independently each particle with probability $q_N$ or by keeping it  with probability $p_N=1-q_N$ and it is determinantal with correlation kernel 
$p_NK^N_V(z,w)$; see the appendix~\ref{A:kernel} for a background overview.  This procedure was introduced in context of random matrix theory  by Bohigas and Pato, \cite{BP04, BP06},  in order to investigate a crossover to Poisson statistics. 
These types of transitions are supposed to arise in various context in statistical physics,   such as the Anderson metal-insulator transition, the crossover from chaotic to regular dynamics in the theory of quantum billiards, or in the spectrum of certain band random matrices, see \cite{Spencer11, EK14a, EK14b} and reference therein.
Although such transitions are believed to be non-universal, the model of Bohigas and Pato is arguably one of the most tractable  to study this phenomenon because it is determinantal. Note that, based on the theory of \cite{HKPV06}, an alternative correlation kernel for this process is given by
\begin{equation} \label{kernel_2}
\widehat{K}^N_V(z,w) = \sum_{k=0}^\infty J_k  \varphi_k(z) \varphi_k(w)
\end{equation}
where  $(J_k)_{k=1}^\infty$ is a sequence of i.i.d. Bernoulli random variables with expected values $\E[J_k]= p_N \1_{k<N} $.  
This shows that removing particles builds up randomness in the system and when the disorder becomes sufficiently strong, it will behave like a Poisson process rather than according to random matrix theory. 
This heuristics applies to general determinantal processes as well and it is straightforward to check that the method presented in section~\ref{sect:cumulant} also applies to the sine or the \Ginibre processes which describes the local limits of the log-gases in dimensions $1$ and $2$ respectively. In fact,  this paper is motivated by an analogous result obtained recently by Berggren and Duits for smooth linear statistics of the incomplete sine and CUE processes, \cite{Berggren17}. 
Based on the fact that these processes come from integrable operators,  they fully characterized the transition for a large class of mesoscopic linear statistics using the corresponding Riemann-Hilbert problem described in~\cite{Deift99}. 
In particular, they obtain analogous crossover for both models and suggested that this should be universal for thinned point processes coming from random matrix theory. 
Prior to~\cite{Berggren17}, they have also been results for the gap probabilities of the critical thinned ensembles. In~\cite{BDIK15, BDIK16}, for the sine process, Deift et al.~computed very detailed asymptotics for the crossover from the {\it Wigner surmise} to the exponential  distribution  making rigorous a prediction of Dyson, \cite{Dyson95}, and Charlier and Claeys obtained an analogous result for the CUE, \cite{CC17}. 
The contribution of this paper is to elaborate on universality
for smooth linear statistics of  $\beta$-ensemble in dimension $1$ or $2$ when $\beta=2$.  Although our proof still relies on the determinantal structure of these models, instead of  the connection with Riemann-Hilbert problems used in the previous works, we apply the cumulants method, which has a combinatorial flavor and appears to be robust to study the asymptotic fluctuations of smooth linear statistics.        \\




To keep the proofs as simple as possible, we will restrict  ourself to real-analytic~$V$, although the results should be valid for more general potential as well (specially in dimension~1 where the asymptotics of the correlation kernels have been studied in great generality). We  keep track of the crossover by looking at linear statistics $\widehat{\Xi}(f)$ of smooth test functions.  
 The random matrix regime is characterized by the property that the fluctuations of the random variable $\widehat{\Xi}(f)$ are of order 1 and described by a {\it universal Gaussian noise} as the number of particles $N\to\infty$. On the other hand, in the Poisson regime, the variance of any non-trivial statistic diverges and, once properly renormalized, the point process converges to a {\it white noise}. 
In the remainder of this introduction, we will formulate our assumptions and limit theorems  for the 2-dimensional log-gases. We refer to section~\ref{sect:1} and~\ref{sect:1'} for  the analogous results in dimension~1. 
The proofs consist in applying the cumulant method introduced by Soshnikov, \cite{Soshnikov00}, to describe the asymptotic laws of linear statistics  of the incomplete process $\widehat\Xi$. 
The details of our strategy are explained in section~\ref{sect:cumulant} and, in particular, they do not depend on the dimension. \\

In what follows, we let $\mathcal{C}^k_0(\S)$ be the set of function with $k$ continuous derivatives and compact support in $\S \subset \X$ and we use the notation:
$$
\p = (\p_x - i \p_y)/2 \ , \hspace{.5cm} \pb = (\p_x + i \p_y)/2 \hspace{.5cm}\text{and}\hspace{.4cm}
\Delta = \p\pb .
$$
If the potential $V$ is real-analytic and satisfies the condition \eqref{potential},
then the log-gas is asymptotically distributed on a compact set $\mathscr{D}_V \subset \C$ which is called the {\it droplet} and its equilibrium density is given by $\varrho_V : = 2\Delta V \1_{\mathscr{D}_V}$ with respect to the area measure $d\A$.  Note that this density may vanish inside the droplet and the {\it bulk} is defined by 
\begin{equation}\label{bulk}
\S_V:= \mathscr{D}_V^\circ \cap \big\{ z\in\C : \Delta V(z) >0 \big\} . 
\end{equation}
%Moreover, the correlation kernel satisfies
%\begin{equation} \label{equilibrium_1}
%K^N_V(z,z) =   2N \Delta V(z) + \frac{1}{2} \Delta\log\big( \Delta V(z)\big) +\O(N^{-1})
%\end{equation}
%uniformly for all $z$ in compact subsets of $\S_V$; see formula (1.12) in \cite{AHM11}.
It is well known that the fluctuations of the eigenvalues of normal random matrices  around the equilibrium configuration are described by a centered Gaussian process $\mathrm{X}$ with correlation structure: 
\begin{equation} \label{H1_noise}
\E\big[\mathrm{X}(f) \mathrm{X}(g) \big] = \frac{1}{4}  \int_\C  \nabla f(z) \cdot \nabla g(z)\ d\A(z) 
=  \int_\C  \p f(z) \pb g(z) d\A(z)    , 
\end{equation}
for any (real-valued) smooth functions $f$ and  $g$.
Modulo constant functions, the RHS~of formula \eqref{H1_noise} defines an Hilbert space, denoted $ H^{1}(\C)$, with norm:
\begin{equation} \label{H1_norm}
\|f \|_{H^1(\C)}^2 =  \int_\C  \left| \p f(z) \right|^2 d\A(z) .    
\end{equation}


  It was proved in \cite{AHM15} that for any smooth function $f$ with compact support, 
\begin{equation} \label{clt_1}
\Xi(f) - \E\big[\Xi(f) \big]  \Rightarrow  \mathrm{X}(f^\dagger)
\end{equation}
as $N\to\infty$, where $f^\dagger$ is the (unique) continuous and bounded function on $\C$ such that  $f^\dagger= f$ on the droplet $\mathscr{D}_V$ and $\Delta f^\dagger =0$ on $\C\backslash\mathscr{D}_V$.  This CLT  was first established for the Ginibre process by Rider and Vir\`{a}g in \cite{RV07a} for $\mathcal{C}^1$ test function with at most exponential growth at infinity. Moreover, if $\supp(f)  \subset \S_V $, then the function $f^\dagger = f$ on $\C$ and the CLT was obtained previously in the paper \cite{AHM11} from which part of the method below is inspired.\\  


We now come to our results. In order  to describe the crossover, let $\Lambda_\eta$ be a mean-zero Poisson process with intensity $\eta \in L^\infty(\X)$. This process is characterized by the fact that for any function $f\in \mathcal{C}_0(\X)$, we have
\begin{equation}\label{Poisson}
\log \E\big[ \exp \Lambda_{\nu}(f) \big] = \int_{\X} \big( e^{f(z)} - 1 - f(z) \big) \eta(z) d\mu(z) . 
\end{equation}

%If $X$ and $Y$ are two real-valued random variables, we denote by $X \oplus Y$ the law of the $X+Y$. 




\begin{theorem} \label{thm:crossover_2}
% Let $\mathrm{X}(f)=  \No\big(0, \| f \|_{H^{\d/2}(\X)}^2\big)   $. 
Let $\mathrm{X}$ be a $H^1$-Gaussian noise and $\Lambda_{\tau\varrho_V}$ be an independent  Poisson process with intensity $\tau\varrho_V$ where $\tau>0$ defined on the same probability space.
Let  $f \in \mathcal{C}^3_0(\S_V)$, $p_N = 1-q_N$, and let  $T_N = N q_N$. 
As $N\to\infty$ and $q_N \to 0$, we have
\begin{align}
\label{Normal_2}
\widehat{\Xi}(f) - \E\big[\widehat{\Xi}(f) \big]  &\Rightarrow  \mathrm{X}(f) &\text{if }\ T_N \to 0 , \hspace{.2cm}\\ 
\label{Poisson_2}
\frac{\widehat{\Xi}(f) - \E\big[\widehat{\Xi}(f) \big]}{\sqrt{T_N} } &  \Rightarrow
 \No\left(0, \int  f(z)^2 \varrho_V(z) d\A(z) \right)   
&\text{if }\ T_N \to \infty ,\\
\label{crossover_2}
\widehat{\Xi}(f) - \E\big[\widehat{\Xi}(f) \big] & \Rightarrow  \mathrm{X}(f) + \Lambda_{\tau\varrho_V}(-f) &\text{if }\ T_N \to \tau  . \hspace{.25cm}
\end{align}
\end{theorem}


The proof of theorem~\ref{thm:crossover_2} relies on the approximation of the correlation kernel $K^N_V$ used in \cite{AHM11}, see lemma~\ref{thm:Bergman_kernel} below, and it restricts us to work with test functions which are supported inside the bulk. However, the result should be true for general functions  if we replace $f$ by $f^\dagger$ on the RHS~of \eqref{Normal_2} and \eqref{crossover_2}.  In fact, for the Ginibre ensemble, $V(z)=|z|^2$, or other nice rotationally invariant potential, one can also apply the combinatorial method of \cite{RV07a}    
to prove the counterpart of theorem~\ref{thm:crossover_2} for any test function $f(z)$ which is a bi-variate polynomial in $z$ and $\bar{z}$. \\

%Theorem~\ref{thm:crossover_2} is proved in section~\ref{sect:2} using the combinatorial framework introduced in section~\ref{sect:cumulant}.  
%This framework also applies to log-gases confined on the real line and the counterpart of theorem~\ref{thm:crossover_2} is stated and proved in section = 

In the regime $Nq_N \to 0$, virtually no particles are deleted and linear statistics behave according to random matrix theory. 
On the other hand, in the regime $Nq_N \to \infty$, the variance of any linear statistic diverges and  the random variable $\hat{\Xi}(f)$ has to be normalized to get a classical CLT. Moreover, formula \eqref{Poisson_2} shows that the normalized process converges to a white noise supported on the  
droplet $\mathscr{D}_V$ whose intensity is  the equilibrium measure~$\varrho_V$. 
In the critical regime, when the expected number of deleted particles equals $\tau>0$, \eqref{crossover_2} shows that the limiting process is the superposition of a $H^1$-correlated Gaussian noise and an independent mean-zero Poisson process  applied to $-f$.  Finally, by using formula \eqref{Poisson}, it is not difficult to check that, as $\tau\to\infty$, the random variable  
$$
\Lambda_{\tau\varrho_V}(-f) / \sqrt{\tau}  \Rightarrow  \No\left(0, \int  f(z)^2 \varrho_V(z) d\A(z) \right)   , 
$$
  so that the critical law interpolates between the regimes \eqref{Normal_2} and~\eqref{Poisson_2}. \\
 
 



The counterpart of theorem~\ref{thm:crossover_2} also holds in the mesoscopic regime. 
The density of the log-gas is of order $N$ and one can also investigate statistics by zooming in the bulk of the process.  Namely, if $L_N \nearrow \infty$,  $x_0 \in \S_V$, and $f\in \mathcal{C}_0(\C)$, we consider the test function
\begin{equation}  \label{meso}
f_N(z) = f\big(L_N(z-x_0) \big) .
\end{equation}



The regime $L_N=N^{1/\d}$ is called microscopic and it was shown in~\cite[Proposition~7.5.1]{AHM11} that 
$$\Xi (f_N) \Rightarrow \Xi^\infty_{\varrho_V(x_0)}(f) , $$
where the process $ \Xi^\infty_{\rho}$ is called the {\bf \Ginibre process} with density $\rho>0$. 
It is a determinantal process on $\C$ with correlation kernel
\begin{equation}\label{Ginibre}
K^\infty_\rho(z,w) =  \rho e^{\rho(2z\bar{w} - |z|^2 -|w|^2 )/2} . 
\end{equation}
Based on the same argument, it is not difficult to verify that the incomplete process has a local limit as well:
\begin{equation}\label{local_limit_1}
\widehat{\Xi} (f_N) \Rightarrow \widehat{\Xi}^{\infty, p}_{\varrho_V(x_0)}(f)
\hspace{.6cm}\text{as}\hspace{.4cm}p_N\to p \hspace{.4cm}\text{and}\hspace{.4cm} N\to\infty. 
\end{equation}
For any $0<p\le 1$, the process $ \hat{\Xi}^{\infty,p}_{\varrho}$ is  determinantal  with correlation kernel  $p K^\infty_\varrho(z,w) $.  This process  is constructed by running an independent Bernoulli percolation with parameter $p$ on the point configuration on  the \Ginibre process with density $\rho>0$. When $p=1$, it coincides with the \Ginibre process and \eqref{local_limit_1} shows that one needs to delete a non-vanishing fraction of the $N$ particles of the gas in order to get a local limit which is different from random matrix theory. 
%
%
It was proved in \cite{RV07b} that, as the density $\rho\to\infty$, the fluctuations of the \Ginibre process are of order 1 and described by the  $H^1$-Gaussian noise:
 $$
 \Xi^\infty_\rho(f) - \rho \int_{\C} f(z) d\A(z)\  \Rightarrow \mathrm{X}(f)
 $$
for any $f \in H^1\cap L^1(\C)$. 
%
%
Therefore it is expected that, in the mesoscopic regime, $\text{i.e. }L_N \prec \sqrt{N}$, the asymptotic fluctuations of the linear statistic  $\Xi(f_N)$ are universal and described by  $\mathrm{X}(f)$.  
However, to our best knowledge, a proof was missing from the literature and, in section~\ref{sect:2}, we show that this fact can be derived by combining the ideas of \cite{RV07b} and \cite{AHM11}. 


\begin{theorem} \label{meso:2}
 Let  $x_0 \in \S_V$, $f \in \mathcal{C}_0^3(\C)$, $\alpha \in (0,1/2)$, and let $f_N$ be given by formula \eqref{meso} with  $L_N =N^\alpha$. Then, the mesoscopic linear statistics
 $$
 \Xi(f_N) - \E\big[ \Xi(f_N)\big] \Rightarrow \mathrm{X}(f)
 $$
 as $N \to\infty$. 
 \end{theorem}

 Using the same method, we can also analyze the fluctuations of smooth mesoscopic linear statistics of the incomplete process.  

\begin{theorem} \label{thm:crossover_3}
% Let $\mathrm{X}(f)=  \No\big(0, \| f \|_{H^{\d/2}(\X)}^2\big)   $.
Let $\mathrm{X}$ be a  $H^1$-Gaussian noise and $\Lambda_{\tau}$ be an independent Poisson process with intensity $\tau>0$ on $\C$.
Let  $x_0 \in \S_V$,  $f \in \mathcal{C}^3_0(\S_V)$, $\alpha \in (0,1/2)$, and let $f_N$ be the mesoscopic test function given by formula \eqref{meso} with  $L_N =N^\alpha$. We also let  $p_N = 1-q_N$ and $T_N= N q_N  L_N^{-2} \varrho_V(x_0)$. We have
\begin{align*}
%\label{Normal_3}
\widehat{\Xi}(f_N) - \E\big[\widehat{\Xi}(f_N) \big]  &\Rightarrow  \mathrm{X}(f) &\text{if }\ T_N \to 0 , \hspace{.2cm}\\ 
%\label{Poisson_3}
\frac{\widehat{\Xi}(f_N) - \E\big[\widehat{\Xi}(f_N) \big]}{\sqrt{T_N} } &  \Rightarrow
 \No\left(0, \int  f(z)^2 d\A(z) \right)   
&\text{if }\ T_N \to \infty ,\\
%\label{crossover_3}
\widehat{\Xi}(f_N) - \E\big[\widehat{\Xi}(f_N) \big] & \Rightarrow  \mathrm{X}(f) + \Lambda_{\tau}(-f) &\text{if }\ T_N \to \tau  ,\hspace{.25cm}
\end{align*}
 as $N\to\infty$ and $q_N\to0$. 
%as $N\to\infty$, where $\tilde{t} =  \varrho_V(x_0) t$. 
\end{theorem}

This result shows that, at mesoscopic scales, the transition occurs when the {\it mesoscopic density of deleted particles} given by the parameter $T_N$ converges to a positive constant $\tau$.
It is  interesting that  we obtain a crossover where the fluctuations are non-Gaussian  and can be described in a simple way. This is to be compared to other analogous transitions for determinantal processes on~$\R$. For instance, one can obtain a crossover from Poisson to GUE eigenvalues by letting  independent points evolved according to  Dyson's Brownian motion. This leads to a determinantal process sometimes called the {\it deformed GUE} whose kernel depends on the diffusion time $\tau$, \cite{Johansson01}. 
The transition for mesoscopic linear statistics has been analyzed in  \cite{DJ16} and  it was proved that 
 the critical fluctuations are Gaussian but the variance depends non-trivially on the test function. One can also consider non-intersecting Brownian motions on a cylinder. It turns out that this point process describes the positions of free fermions confined in a harmonic trap at a temperature $\tau>0$. 
 It was established in \cite{Johansson07}, see also \cite{DDMS14}, that the corresponding grand canonical ensemble is determinantal with a correlation kernel of the form \eqref{kernel_2} with  for all $k\ge 0$, 
 $$\E[J_k] = \frac{1}{1+ \exp(\frac{k-N}{\tau})} . $$  
 For sufficiently small temperature, this system behaves like its ground-state, the GUE, while it
behaves like a Poisson process at larger temperature.
 It was proved in \cite{JL15} that this leads to yet another crossover where non-Gaussian fluctuations are observe at the critical temperature. However,  to the author's knowledge, in contrast to the incomplete ensembles considered in this note, the critical processes discovered  in \cite{JL15}  cannot be described in simple terms like~\eqref{crossover_2}. \\
 
 

 
 
 
%  and it leads to another crossover mechanism. 
 
 %where $t$ is the temperature and  it behaves like its ground-state, the GUE, at small temperature and like a Poisson process at larger temperature.  
 





The rest of the paper is organized as follows. In section~\ref{sect:cumulant}, we review Soshnikov's cumulants method and how to apply it to the incomplete ensemble $\hat{\Xi}$.  In section~\ref{sect:2}, we prove theorems~\ref{thm:crossover_2}--\ref{thm:crossover_3}  for the 2-dimensional log-gases. 
The proof relies on estimates for the correlation kernel \eqref{kernel} which come from the paper~\cite{AHM11} and are collected in the appendix~\ref{A:damping}. 
The counterparts  of  theorem~\ref{thm:crossover_2} and theorem~\ref{thm:crossover_3} in dimension $1$ are stated and proved in section~\ref{sect:1} and~\ref{sect:1'} respectively.  
In the global regime, we give a combinatorial proof which is based on the asymptotics of the recurrence coefficients of the orthogonal polynomials \eqref{OP}. This argument can be generalized to other  biorthogonal ensembles; see remark~\ref{rk:biorthogonal}. 
In the mesoscopic regime, the analysis is similar to the two-dimensional case and relies on the asymptotics of the correlation kernel $K^N_V$ and the techniques introduced in~\cite{Lambert_a}. Finally, $C>0$ denotes a numerical constant which changes from line to line and, for any $n\in\N$, we let for all $\mathrm{x} \in\X^n$, 
$$
 d\mu^n(\mathrm{x}) = d\mu(x_1) \cdots d\mu(x_n) . 
$$



%Since the process has a correlation kernel $p_NK^N_V(x,y)$, for any nice test function,


  
  \subsection*{Acknowledgements}
  I thank Tomas Berggren and Maurice Duits from which I learned about the model of Bohigas and Pato,   for sharing their inspiring work with me and for the valuable discussions which followed. I also thank Mariusz Hynek for reading this draft and for many interesting discussions.
  
  \section{Outline of the proof} \label{sect:cumulant}

   If $(\mathrm{u}_N)$ and $(\mathrm{v}_N)$ are two sequences, we will use the notation:  $$\mathrm{u}_N \simeq \mathrm{v}_N 
  \hspace{.5cm}\text{if} \hspace{.5cm} \lim_{N\to\infty}( \mathrm{u}_N - \mathrm{v}_N) =0  
  $$  
  and 
$$
\mathrm{u}_N \prec \mathrm{v}_N 
\hspace{.5cm}\text{if} \hspace{.5cm} \lim_{N\to\infty} \mathrm{u}_N / \mathrm{v}_N =0  . 
  $$
  Suppose that $\Xi$ is a determinantal point process on $\X$ with a correlation kernel $K\in L^2(\X\times \X, \mu\times \mu)$ which is reproducing:
\begin{equation} \label{projection}
\int_\X K(z, x) K(x, w) d\mu(x) = K(z,w)
\end{equation}
 for all $z,w \in \X$. Since it is the case for most ensembles, we shall also assume that the kernel  $K(z,w)$ is continuous  on $\X\times \X$. 
 %
 % 
 The cumulants method  to analyze the asymptotic distribution of linear statistic of determinantal processes goes back to the work of  Costin and Lebowitz,  \cite{CL95},  for count statistics of  the sine process and the general theory was developed by Soshnikov,  \cite{Soshnikov, Soshnikov00, Soshnikov01},  and subsequently applied to many different ensembles coming from random matrix theory, see for instance \cite{RV07a, RV07b, AHM11, BD16, BD17, JL15, Lambert_a, Lambert_b}.
%
  %  
%Then this method has been applied to analyze the  asymptotic fluctuations of various ensembles coming from random matrix theory or other probabilistic models, %
%
 In this section, we show how to implement it to describe the asymptotics law of linear statistics of the incomplete ensemble $\hat{\Xi}$ with correlation kernel $p K(z,w)$ when  $0<p<1$. \\

   
Let 
  $$
  \mho= \bigcup_{l=1}^\infty \big\{ \k=(k_1,\dots, k_l) \in \N^l  \big\} 
  $$
  and let $\ell(\k)= l$ denote the length of $\k$. Then, the set of compositions of the integer $n>0$ is 
  $$ \big\{ \k \vdash n \big\} = \big\{ \k\in\mho :  k_1+ \cdots + k_l = n \} . $$ 
    %$ if $|\k| = k_1+ \cdots + k_l = n $. 
  % In particular,  $\big\{\k \in   \mho : \k \vdash n   \big\}$ is the set of compositions of the integer $n\in\N$.
We denote by $n\in \mho$ the trivial composition. 
  For any map $\Upsilon : \mho \mapsto \R $, for any function $f: \X \to \R$, and for any $n\in\N$, we define   for all $x\in \X^n$,
\begin{equation}\label{Upsilon_f}
  \Upsilon^n[f](x) = \sum_{\k \vdash n} \Upsilon(\k)\hspace{-.2cm} \prod_{1 \le j\le \ell(\k)} \hspace{-.2cm}f(x_j)^{k_j} . 
 \end{equation}
If $\k \vdash n$, we let $\displaystyle \M(\k) =\frac{n!}{k_1!\cdots k_l!}$ be the multinomial coefficient and for all integers $n\ge 1$ and $ m \in\{0, \dots , n\}$, we define the coefficients 
\begin{equation}\label{Gamma}
  \gamma^n_m = \sum_{ \k \vdash n}\frac{(-1)^{\ell(\k)}}{\ell(\k)} {\ell(\k) \choose m} \M(\k) .  
\end{equation}

We will also use the notation:
$\displaystyle\delta_{k}(n) = \begin{cases} 1 &\text{if}\ n=k \\ 0 &\text{else} \end{cases}$ for any $k\in\mathbb{Z}$.


      \begin{lemma} \label{thm:combinatorics}
  For all $n\in \N$, we have $\gamma^n_0 = \delta_{1}(n)$ and $  \gamma^n_1 = (-1)^n$.
  \end{lemma}
\proof  The coefficients \eqref{Gamma}  have the generating function:
  $$
  \sum_{n=1}^\infty \sum_{m=0}^n   \gamma^n_m  \frac{x^n q^m}{n!}  = - \log\big(1+(1+ q)(e^x-1)\big) . 
  $$
  In particular, setting $q=0$, we see that $  \gamma^n_0 = \delta_{1}(n) $. 
Moreover, since
 $$
 \sum_{n=1}^\infty   \gamma^n_1  \frac{x^n}{n!}    
 =\left.  \frac{d}{dq}\log\big(1+(1+ q)(e^x-1)\big) \right|_{q=0} =1-e^{-x} , 
 $$
  we also see that $  \gamma^n_1 = (-1)^n$ .\qed\\ 
  
  


  
  
  
  
Given a test function  $f: \X \to \R$, the cumulant generating function of the random variable $\Xi(f)$ is 
$$
\log \E\big[ \exp\{ \lambda \Xi(f)\} \big] = \sum_{n=1}^\infty \frac{\lambda^n}{n!} \Cu^n_{K}[f] .
$$
  It is known that under general  assumptions, the cumulants $\Cu^n_{K}[f] $ characterized the law of the linear statistics  $\Xi(f)$ and it was proved by Soshnikov that for any $n\in\N$, 
\begin{equation} \label{Soshnikov}
\Cu^n_{K}[f]
= - \sum_{l=1}^n   \frac{(-1)^l}{l} \sum_{\begin{subarray}{c} \k \vdash n \\ \ell(\k) =l \end{subarray}} \M(\k) 
\underset{x_{0} =x_l}{\int_{\X^l}} f(x_1)^{k_1} \cdots  f(x_l)^{k_l} \prod_{1\le j\le l} K(x_j, x_{j-1}) d\mu^l(x) ,  
\end{equation}
whenever this formula makes sense. Note that we use the convention that the variables $x_0$ and $x_l$ are identified in the previous integral. If the correlation kernel $K$ is reproducing, then we can rewrite this formula:
\begin{align} \notag
\Cu^n_{K}[f]
&= - \sum_{\k \vdash n}   \frac{(-1)^{\ell(\k)}}{\ell(\k)} \M(\k) 
\underset{x_{0} =x_n}{\int_{\X^n}} f(x_1)^{k_1} \cdots  f(x_l)^{k_l} \prod_{1\le j\le n} K(x_j, x_{j-1}) d\mu^n(x) \\
&\label{cumulant_0}
= -\underset{x_{0} =x_n}{\int_{\X^n}}\Upsilon^n_0[f](x)  \prod_{1\le j\le n} K(x_j, x_{j-1}) d\mu^n(x)  ,
\end{align}
where for any $\k\in\mho$,
\begin{equation}\label{Upsilon_0}
\Upsilon_0(\k) = \frac{(-1)^{\ell(\k)}}{\ell(\k)} \M(\k)  . 
\end{equation}

A simple observation which turns out to be very important when it comes to the asymptotic analysis is  that, by lemma~\ref{thm:combinatorics}, for all $n\ge 2$,
\begin{equation} \label{normalization_0}
\sum_{\k \vdash n} \Upsilon_0(\k)  =\gamma_0^n =0 . 
\end{equation}




For any $m\in\N$, we define
\begin{equation} \label{Upsilon}
  \begin{cases}
\displaystyle  \Upsilon_m(\k) = \frac{(-1)^{\ell(\k)}}{\ell(\k)} {\ell(\k) \choose m} \M(\k)  & \text{if } \ell(\k) \ge 2 \\
\displaystyle  \Upsilon_m(\k) =- \delta_{1}(m)-\gamma^n_m  & \text{if } \k=n
  \end{cases} .
  \end{equation}
  By convention $ {l \choose m} = 0 $ if $m>l$ and these functions are constructed so that we have for all $n,m\in\N$, 
\begin{equation} \label{normalization}
 \sum_{\k \vdash n} \Upsilon_m(\k) = 0 .
 \end{equation}


 
  
Let $0<q<1$ and $p=1-q$. According to formula \eqref{Soshnikov}, the cumulants of the process $\hat{\Xi}$ with correlation kernel $\widehat{K} = p K$ are given by 
  \begin{equation*}
\Cu^n_{\widehat{K}}[f]
= % \sum_{\k \vdash n}   \frac{(-1)^{\ell(\k)}}{\ell(\k)} \M(\k) 
- \sum_{l=1}^n   \frac{(-1)^l}{l} p^l \sum_{\begin{subarray}{c} \k \vdash n \\ \ell(\k) =l \end{subarray}} \M(\k)
\underset{x_{0} =x_l}{\int_{\X^l}} f(x_1)^{k_1} \cdots  f(x_l)^{k_l} \prod_{1\le j\le l} K(x_j, x_{j-1}) d\mu^l(x) 
\end{equation*}
  and since the kernel $K$ is reproducing,  using the binomial formula, we obtain
    \begin{align}  \label{cumulant}
\Cu^n_{\widehat{K}}[f]
= \Cu^n_{K}[f]  %& + q  \bigg\{ \gamma^n_1 \int_\X  f(x)^{n} K(x,x) d\mu(x)  +  \underset{x_{0} =x_n}{\int_{\X^n}}     \Upsilon^n_1[f](x)  \prod_{j\le n} K(x_j, x_{j-1}) d\mu^n(x)  \bigg\}\\
&- \sum_{m=1}^n (-q)^m  \gamma^n_m \int_\X  f(x)^{n} K(x,x) d\mu(x)  \\
&\notag-    \sum_{m=1}^n (-q)^m  \underset{x_{0} =x_n}{\int_{\X^n}}     \Upsilon^n_m[f](x)  \prod_{1\le j\le n} K(x_j, x_{j-1}) d\mu^n(x) .
\end{align}
  

Now, let $(K^N)_{N\in\N}$ be a sequence of reproducing kernels and suppose that there exists a probability measure $\mueq$ so that for any $n\in\N$, 
\begin{equation} \label{LLN_1}
 \int_\X f(x)^n  K^N(x,x) d\mu(x) \simeq N  \int_\X f(x)^n d\mueq(x). 
\end{equation}
 We also assume that there exists $\kappa>0$ such that  for all $n\ge 2$ and all  $m\ge 1$,
\begin{equation}\label{assumption_1}
 \left| \underset{x_{0} =x_n}{\int_{\X^n}}     \Upsilon^n_m[f](x)  \prod_{1\le j\le n} K^N(x_j, x_{j-1}) d\mu^n(x) \right| \prec (\log N)^\kappa . 
\end{equation}
%The error terms in \eqref{LLN_1} and \eqref{assumption_1} may depend on the test function~$f$ and the parameters $n, m$. 
In the case of the log-gases, formula \eqref{LLN_1} follows either from the asymptotics of the correlation kernel \eqref{kernel} or  from potential theory and $\mueq$   is called the equilibrium measure.  
Moreover, under the assumption \eqref{potential}, this measure is absolutely continuous and it  has compact support. 
The main technical challenge of this paper is to verify that  the estimate \eqref{assumption_1} holds for a large class of test functions.
In fact, if $f$ is smooth, we expect these integrals to be uniformly bounded in $N$ for general determinantal processes but a logarithmic error is allowed for our applications. In the remainder of this section, we will show how these assumptions and the CLT for the classical log-gas imply theorem~\ref{thm:crossover_2}.
The proofs  of theorem~\ref{thm:crossover_3},  as well as theorems~\ref{thm:crossover_1} and~\ref{thm:crossover_4} in  the 1-dimensional  setting, all follow the same strategy and the details are given in sections~\ref{sect:2},~\ref{sect:1} and~\ref{sect:1'}  respectively. \\
%In particular, in dimension 2, the asymptotics \eqref{LLN_2} implies that  \eqref{LLN_1} holds for any $f\in \mathcal{C}_0(\S_V)$  and  $\rhoeq = \Delta V \1_{}$ 




For a 2-dimensional log-gas, the equilibrium measure satisfies $d\mueq= \varrho_V d\A$ where $\varrho_V = 2 \Delta V \1_{\mathscr{D}_V}$ and,  by lemma~\ref{thm:Bergman_kernel} below, formula \eqref{LLN_1} holds for any test function $f\in \mathcal{C}_0(\S_V)$. Moreover, it was also proved in~\cite[Theorem~4.4]{AHM11} that if $f\in \mathcal{C}_0^3(\S_V)$, then
\begin{equation} \label{CLT_0}
\lim_{N\to\infty} \Cu^n_{K^N_V}[f] = \begin{cases} \|f \|_{H^1(\C)}^2 &\text{if } n=2 \\ 0 &\text{if } n>2 
\end{cases} .
\end{equation}




Let $0<q_N<1$ be a sequence which converges to 0 as $N\to\infty$, $p_N = 1- q_N$ and let $\widehat{K}^N_V = p_NK^N_V$.   By formula \eqref{cumulant}, for any function $f\in \mathcal{C}_0^3(\S_V)$ which satisfies the condition~\eqref{assumption_1}, we obtain
\begin{equation} \label{limit_1}
\Cu^n_{\widehat{K}^N_V}[f]=\Cu^n_{K^N_V}[f] + T_N \int_\C  f(z)^{n}  \varrho_V(z) d\A(z)  
+ \underset{N\to\infty}{o}(T_N)  ,
 \end{equation}
where $T_N  =  Nq_N$.
In the regime where $T_N \to 0$,  formula \eqref{limit_1} implies that $\Cu^n_{\widehat{K}^N_V}[f] \simeq \Cu^n_{K^N_V}[f]$ for all $n\ge 2$. Hence, once centered, both statistics $\Xi(f)$ and $\hat{\Xi}(f)$ converge to a Gaussian random  variable with variance $\|f \|_{H^1(\C)}^2$.  \\


On the other hand, in the regime where $T_N \to \infty$, if we let  $h_N := f/\sqrt{T_N} $, by rescaling formula \eqref{limit_1} we obtain
\begin{equation*} 
\Cu^n_{\widehat{K}^N_V}[h_N] =  T_N^{1-n/2}\int_\C  f(x)^{n}  \varrho_V(z) d\A(z)+ \O\big(T_N^{-n/2} \big)  .
 \end{equation*}
Just like for a Poisson point process with intensity function $T_N \varrho_V$, this implies that $\hat{\Xi}(h_N)- \E\big[\hat{\Xi}(h_N)\big]$ converges in distribution to a Gaussian random variable with variance $\displaystyle \int_\C  f(x)^{2}  \varrho_V(z) d\A(z)$.\\
% and we get \eqref{Poisson_2}. \\

 Finally in the critical case where $T_N \to \tau$,   formulae \eqref{limit_1} and \eqref{CLT_0} implies that
\begin{equation} \label{limit_2}
\lim_{N\to\infty}\Cu^n_{\widehat{K}^N_V}[f] =  \delta_2(n) \|f \|_{H^1(\C)}^2 + \tau  \gamma^n_1 \int_\C  f(x)^{n}  \varrho_V(z) d\A(z) . 
 \end{equation}

By formula \eqref{Poisson}, the cumulants of the random variable  $\Lambda_{\eta}(f)$
 are given by
 $$
 \Cu^n\big( \Lambda_{\eta}(f)\big) =    \int_{\X}   f(z)^{n} \eta(z)  d\mu(z) 
 \hspace{.5cm}\text{for all}\hspace{.3cm} n\in \N ,
 $$
and by lemma~\ref{thm:combinatorics}, the coefficient $\gamma^n_1= (-1)^n$. Hence, formula \eqref{limit_2} implies that  the random variable
$ \hat{\Xi}(f) -\E\big[ \hat{\Xi}(f)\big] $ converges in distribution  to the sum of a Gaussian random variable
with variance $\|f \|_{H^1(\C)}^2$ and an independent  random variable $\Lambda_{t \varrho_V}(-f)$.  \\


\section{Transition for the log-gases in dimension 2}\label{sect:2}

We begin by reviewing the basics of the theory of eigenvalues of random normal matrices developed  by Ameur, Hedenmalm and Makarov. In particular, we are interested in the properties of the correlation kernel \eqref{kernel} in the bulk of the gas.  Recall that the equilibrium measure  is $\varrho_V  = 2\Delta V \1_{\mathscr{D}_V}$, the droplet $\mathscr{D}_V$ is a compact set with a  nice boundary, and the set $ \S_V$ is given by \eqref{bulk}. 
% We let $$ \S_V := \big\{z\in\C : \Delta V (z) >0 \big\} \cap \mathscr{D}_V^\circ .$$
%
%
It was pointed out in \cite{AHM11} that, if the potential $V$ is real-analytic, in order to compute the asymptotics of the cumulants of a smooth linear statistic, instead of working with  the correlation kernel $K_N^V$, one can use the so-called approximate Bergman kernel,
 \begin{equation}\label{B_kernel}
B^N_V(z,w) = \big( N b_0(z, \bar{w}) + b_1(z, \bar{w}) \big) e^{N \{ 2 \Phi(z, \bar{w}) - V(z) - V(w) \} } . 
\end{equation}
The functions  $b_0(z,w)$, $b_1(z,w)$ and $\Phi(z,w)$ are the (unique) bi-holomorphic functions defined in a neighborhood  in $\C^2$ of the set $\big\{ (z ,\bar{z}) : z\in \S_V \big\}$  such that 
$b_0(z,\bar{z}) = 2 \Delta V(z)$, $b_1(z,\bar{z}) = \frac{1}{2} \Delta \log( \Delta V)(z)$, and $\Phi(z,\bar{z}) = V(z)$. 

 
 
 
\begin{lemma}[Lemma~1.2 in \cite{AHM11}, see also \cite{Berman09, AHM10}] \label{thm:Bergman_kernel}
For any $x_0 \in \S_V $, there exists $\epsilon_0>0$ and $C>0$ so that
\begin{equation} \label{approximation_1}
\left| K^N_V(z, w) - B^N_V(z,w) \right| \le C N^{-1}
\end{equation}
for all $z, w  \in \D(x_0 , \epsilon_0)$. 
\end{lemma} 
 
Moreover, at sufficiently small mesoscopic scale, up to a gauge transform, the asymptotics of the  approximate Bergman kernel $B^N_V$ is universal.

\begin{lemma} \label{thm:Ginibre_approximation} 
Let $\kappa>0$ and  $\epsilon_N= \kappa N^{-1/2}\log N $ for all $N\in \N$.
For any  $x_0 \in \S_V$, there exists $\varepsilon_0>0$ and  a function $\h : \D(0,\varepsilon_0)  \to \R$ such that if  the parameter $N$ is sufficiently large, the function 
\begin{equation} \label{B}
\widetilde{B}^N_{V, x_0}( u ,v)  =\frac{B^N_V(x_0 + u , x_0 + v)e^{i N \h(u)}}{e^{i N \h(v)}} 
\end{equation} 
satisfies
\begin{equation} \label{asymptotics_B}
\widetilde{B}^N_{V, x_0}( u , v) = K^\infty_{N \varrho_V(x_0)}(u, v)\left\{ 1 + \underset{N\to\infty}{O}\big(  (\log N)^{2}\epsilon_N   \big)  \right\}
\end{equation}
uniformly for all $u, v \in \D(0, \epsilon_N)$, where $K^\infty_{N \varrho_V(x_0)}$ is the \Ginibre kernel with density $N \varrho_V(x_0)= 2N \Delta V(x_0)$. 
\end{lemma}

A key ingredient in the paper \cite{AHM11}, as well as \cite{RV07b},  is to reduce the domain of integration in formula \eqref{cumulant_0}, using the exponential  off-diagonal decay of the correlation kernels $K_V^N$, to a set where we can use the asymptotics \eqref{asymptotics_B} and approximate the function $\Upsilon^n_0[f]$ by a multivariate polynomial of degree 2 using Taylor's theorem.
This reduction step is formulated in the following lemma.
For completeness, the proofs of lemmas~\ref{thm:Ginibre_approximation} and~\ref{thm:localization} are given in the appendix~\ref{A:damping}. 

\begin{lemma} \label{thm:localization}
Let $n\in\N$ and $\epsilon_N = \kappa N^{-1/2} \log N$ for some constant $\kappa>0$ which is sufficiently large compared to $n$. We let
 \begin{equation} \label{A}
\mathscr{A}(z_0; \epsilon)
= \big\{ \z\in\C^n  :  |z_j - z_{j-1}| \le \epsilon \text{ for all } j=1, \dots n  \big\} .  
\end{equation}
Let $\S$ be a compact subset of $\S_V$, $N_0\in\N$,  and $F_N : \C^{n+1} \to \R $ be a sequence of  continuous functions such that 
\begin{equation}\label{estimate_0}
 \sup\big\{ |F_N(z_0, \z)| : \z \in \C^{n} , N  \ge N_0 \big\}  \le C \1_{z_0 \in\S} .
 \end{equation}
We have
\begin{align}
&\label{localization} \underset{z_0 = z_{n+1}}{\int_{\C^{n+1}}}\hspace{-.2cm} F_N(z_0, \z) \prod_{j=0}^n K^N_V(z_{j}, z_{j+1}) d\A(z_0) d\A^{n}(z)  \\
&\notag\hspace{1cm}
= \int_\S d\A(z_{0})  \underset{z_{n+1} = z_{0}}{\int_{\mathscr{A}(z_{0}; \epsilon_N)}}\hspace{-.2cm} F_N(z_0,\z)  \prod_{j=0}^n K^N_V(z_{j}, z_{j+1}) d\A^{n}(\z)\ +  \O(N^{-1}) .
\end{align}
\end{lemma}

Using the fact that 
\begin{equation} \label{Ginibre_bound}
\big| K^\infty_\rho(z,w)\big| = \rho e^{-\rho |z-w|^2/2} 
\end{equation}
for all $z, w\in\C$ and $\rho>0$, it is easy to obtain a similar estimate for the \Ginibre kernel, see lemma~\ref{thm:delocalization}. 
The next proposition is the technical step to prove that the 2-dimensional log-gases satisfy the assumption \eqref{assumption_1}. 


\begin{proposition} \label{thm:approximation}
Let $\K\subset \C$ be a compact set, $f\in \mathcal{C}^3_0(\K)$ and let $\Upsilon: \mho \to\R$ be any map such that  $\sum_{\k \vdash n} \Upsilon(\k) = 0$ for all $n\ge2$. Let $L_N$ be an increasing sequence such that $1\prec L_N \prec  \sqrt{N}/(\log N)^{3}$,  $x_0 \in \S_V$ and $f_N=f(L_N(z- x_0))$. We also denote by $H^n(\lambda; \w )$  the second order Taylor polynomial at $0$ of the function 
$\w\in\C^n \mapsto   \Upsilon^{n+1}[f](\lambda, \lambda+ \w)$. 
Then, we have for any  $n\ge 1$,
\begin{align} \label{approximation}
& \underset{z_0 = z_{n+1}}{\int_{\C^{n+1}}}\hspace{-.2cm}  \Upsilon^{n+1}[f_N](z_0, \z) \prod_{j=0}^n K^N_V(z_{j}, z_{j+1}) d\A(z_0) d\A^{n}(\z)  \\
&\notag
= \int_{\mathcal{K}} d\A(\lambda)
\hspace{-.3cm}\underset{w_0=w_{n+1}=0}{\int_{\C^n}}\hspace{-.4cm}
H^n(\lambda; \w)
\prod_{j=0}^{n} K^\infty_{\eta_N(\lambda)}(w_{j}, w_{j+1}) \  d\A^n(\w)  
+\O\left( \frac{(\log N)^4}{L_N} \right) , 
\end{align}
where the density is given by $\eta_N(\lambda) = N L_N^{-2} \varrho_V(x_0+ \lambda/L_N)$. %In particular, note that by assumptions $\eta_N(\lambda) \to\infty$ for all $\lambda\in\S_0$. 
\end{proposition}

\begin{remark} \label{rk:macro}
In the macroscopic regime ($L_N=1$ and $x_0=0$),  the proof below
shows that the asymptotics \eqref{approximation} still holds as long as $\K \subset\S_V$.
It is also worth noting that the error term in formula \eqref{approximation} is not optimal and it is expected to converge to 0 for any scale   $1\le L_N \prec \sqrt{N}$.
% In the macroscopic regime,  this has been proved in~\cite{AHM11} when $\Upsilon=\Upsilon_0$.
\end{remark}



\proof We let $F_N = \Upsilon^{n+1}[f_N]$ and 
$$\mathscr{J}^{n}_N : =  \underset{z_{n+1}=z_0}{\int_{\C^{n+1}}}\hspace{-.2cm} F_N(z_0, \z) \prod_{j=0}^n K^N_V(z_{j}, z_{j+1}) d\A(z_0) d\A^{n}(\z)  . $$
%Note that we abuse of notation in the previous formula and denoted $F_N( \z) = F_N(z_0, z_1,\dots, z_n)$. 
Since $x_0\in\S_V$, there exists a compact set $\S\subset \S_V$ so that $\supp(f_N) \subseteq \S$ when the parameter $N$ is sufficiently large and, according to formula \eqref{Upsilon_f}, the function $F_N$ satisfies the assumption \eqref{estimate_0}.   By   lemma~\ref{thm:localization}, this implies that
\begin{equation} \label{cumulant_1}
\mathscr{J}^{n}_N \simeq \int_{\S} d\A(z_0)  \underset{z_{n+1} = z_0}{\int_{\mathscr{A}(z_0; \epsilon_N)}}\hspace{-.3cm} F_N(z_0,\z)  \prod_{j=0}^n K^N_V(z_{j}, z_{j+1}) d\A^{n}(\z) . 
\end{equation}  

By \eqref{A},
\begin{equation*}
\mathscr{A}(z_0; \epsilon)
\subset \big\{ \z\in\C^n :   z_1, \dots, z_n \in  \D(z_0,n \epsilon_N) \big\} ,
\end{equation*}
and we can apply lemma~\ref{thm:Bergman_kernel} to replace the kernels $K^N_V(z_j,z_{j+1})$ in formula \eqref{cumulant_1}.
 Namely, if $z_{n+1} = z_0 $ and $\z \in \mathscr{A}(z_0; \epsilon)$, then
 %Namely for any $z_0 \in \S_0$ and for all $\z \in \mathscr{A}(z_0;\epsilon_N)$,  if  $z_{n+1} = z_0$, then
\begin{equation*}
\Bigg| \prod_{j=0}^n K^N_V(z_{j}, z_{j+1}) - 
 \prod_{j=0}^{n} B^N_V(z_{j}, z_{j+1}) \Bigg| \le C \sum_{k=1}^{n+1} N^{-k} S_N^{n+1-k} ,  
\end{equation*}
where $S_N =  \sup\big\{ |B^N_V(z,w)|  :  z, w \in \D(z_0, n \epsilon_N)  , z_0 \in \S  \big\} $. 
By  lemma~\ref{thm:Ginibre_approximation}, for  all  $u,v \in \D(0, n\epsilon_N) $, 
$$
\big| B^N_V(z_0 + u , z_0 + v) \big| \le C
\big|  K^\infty_{N\varrho_V(z_0)}(u,v) \big| .
$$
and, by formula \eqref{Ginibre_bound}, this implies that $S_N  \le  C N$.
If we combine these estimates with formula \eqref{cumulant_1}, since the function $F_N$ are uniformly bounded, we obtain
\begin{align*} 
\mathscr{J}^{n}_N
=  \int_{\S} d\A(z_0)  \underset{z_{n+1} = z_0}{\int_{\mathscr{A}(z_0; \epsilon_N)}}\hspace{-.3cm} F_N(z_0,\z)  \prod_{j=0}^n B^N_V(z_{j}, z_{j+1}) d\A^{n}(\z)\\
\hspace{3cm}+\O\left(  N^{n-1}  \int_{\S} d\A(z_0)  \big|\mathscr{A}(z_0; \epsilon_N)\big| \right)
\end{align*}  
By definition, $\epsilon_N = \kappa N^{-1/2} \log N$ so that 
$ \big| \mathscr{A}(z_0;\epsilon_N)  \big|  \le C N^{-n} (\log N)^{2n}$ 
for all $z_0\in \C$. Thus, the previous error term converges to 0 like $(\log N)^{2n}/ N$.
 Hence, if we make the change of variables $\z = z_0 +\u$ and the appropriate {\it gauge transform} in the previous integral, according to formula \eqref{B}, we obtain
\begin{equation}  \label{cumulant_2}
\mathscr{J}^{n}_N
\simeq  \int_{\S} d\A(z_0)  \underset{u_{n+1} = u_0=0}{\int_{\mathscr{A}(0; \epsilon_N)}}\hspace{-.3cm} F_N(z_0 +\u) \prod_{j=0}^n \widetilde{B}^N_{V, z_0}(u_{j}, u_{j+1}) d\A^{n}(\u) . 
\end{equation}
Note that in formula \eqref{cumulant_2}, the integral  is over a small subset of the surface $\{ \u \in \C^{n+2} : u_0 = u_{n+1} =0\} $ and we denote 
$F_N(z_0 +\u) = F_N(z_0, z_0+u_1,\dots, z_0+u_n)$. 
%We will use this convention in the rest of the proof.
Then, we  can apply lemma~\ref{thm:Ginibre_approximation} to replace the kernel $ \widetilde{B}^N_{V, z_0}$ by its local limit  $K^\infty_{N \varrho_V(z_0)}$ in formula \eqref{cumulant_2}, we obtain 
\begin{equation*}  
\mathscr{J}^{n}_N
\simeq  \int_{\S} d\A(z_0)  \underset{u_{n+1} = u_0=0}{\int_{\mathscr{A}(0; \epsilon_N)}}\hspace{-.3cm} F_N(z_0 +\u)\chi_N(z_0,\u)  \prod_{j=0}^n K^\infty_{N \varrho_V(z_0)}(u_{j}, u_{j+1})  d\A^{n}(\u)  ,
\end{equation*}
where $\displaystyle \chi_N(z_0,\u) = 1 + \underset{N\to\infty}{O}\big(  (\log N)^{2}\epsilon_N   \big)$ uniformly for all $z_0\in\S$ and all $\u\in \mathscr{A}(0; \epsilon_N)$. \\

Let $F=\Upsilon^{n+1}[f]$, $\delta_N = \epsilon_N L_N$ and $\eta_N(\lambda) =N L_N^{-2} \varrho_V(x_0 + \lambda/L_N) $. By definition, 
$F_N(z_0 +\u) = F\big(L_N(z_0- x_0 +\u )\big) $ and 
we can make the change of variables $\lambda = L_N(z_0-x_0)$ and $\w= L_N \u$ to get rid of the scale $L_N$ and the fixed point $x_0$ in the previous integral. Using the obvious scaling property of the \Ginibre kernel, \eqref{Ginibre}, we obtain
\begin{equation} \label{cumulant_3}  
\mathscr{J}^{n}_N
\simeq  \int_{\K} d\A(\lambda)  \underset{w_{n+1} = w_0=0}{\int_{\mathscr{A}(0; \delta_N)}}\hspace{-.3cm} F(\lambda+\w)\tilde{\chi}_N(\lambda,\w)  \prod_{j=0}^n K^\infty_{\eta_N(\lambda)}(w_{j}, w_{j+1})  d\A^{n}(\w) ,  
\end{equation}
where $\displaystyle \tilde{\chi}_N(\lambda,\w)= 1 + \underset{N\to\infty}{O}\big(  (\log N)^{2}\epsilon_N   \big)$ uniformly for all $\lambda\in\K$ and for all $\u\in \mathscr{A}(0; \delta_N)$. 
The condition $\sum_{\k \vdash n+1} \Upsilon(\k) = 0$ implies that $F(\lambda+0)=0$ for all $\lambda \in \C$ so that  
$$ \big| F(\lambda+\w) \big| \le C \delta_N$$
for all $\lambda\in \K$ and all $\w\in \mathscr{A}(0; \delta_N)$. 
Moreover, by formula~\eqref{Ginibre_bound},  for any $n \in \N$, 
\begin{equation} \label{estimate_5}
\prod_{j=0}^{n}  \big|K^\infty_{\rho}(w_{j}, w_{j+1})\big| \le \rho^{n+1}   \prod_{j=1}^{n}  e^{- \rho | v_j |^2 /2 } 
\end{equation}
where $v_j = w_j - w_{j-1}$ for all $j=1,\dots, n$. Hence, we see that
$$
\bigg| \underset{w_{n+1} = w_0=0}{\int_{\mathscr{A}(0; \delta_N)}}\hspace{-.3cm} F(\lambda+\w) \prod_{j=0}^n K^\infty_{\eta_N(\lambda)}(w_{j}, w_{j+1})  d\A^{n}(\w)  \bigg| \le C \delta_N \eta_N(\lambda) 
$$
and, since $\eta_N(\lambda) \le C N L_N^{-2}$ for all $\lambda\in \K$, we deduce from formula \eqref{cumulant_3} that
\begin{align}  \label{cumulant_4}  
\mathscr{J}^{n}_N
= \int_{\K} d\A(\lambda)  \underset{w_{n+1} = w_0=0}{\int_{\mathscr{A}(0; \delta_N)}}\hspace{-.3cm} F(\lambda+\w) \prod_{j=0}^n K^\infty_{\eta_N(\lambda)}(w_{j}, w_{j+1})  d\A^{n}(\w)  \\
+ \notag\O\big( N L_N^{-2} \delta_N \epsilon_N (\log N)^2 \big) .
\end{align}
Recall that $\delta_N= L_N \epsilon_N$ and $\epsilon_N=\kappa N^{-1/2}\log N$, so that the error term in \eqref{cumulant_4} is of order $(\log N)^4 L_N^{-1}$. 
Moreover,  if  $L_N \prec \sqrt{N}/\log N$,   a Taylor approximation shows that for any $\w \in \mathscr{A}(0; \delta_N) $, 
\begin{equation*} \label{Taylor}
F(\lambda,\lambda+w_1,\dots,  \lambda+w_n)  =H^n( \lambda; \w) + \O(\delta_N^3) . 
\end{equation*}
Using the estimate \eqref{estimate_5} once more, by formula \eqref{cumulant_4}, this implies that
\begin{align}  \label{cumulant_5}  
\mathscr{J}^{n}_N
=  \int_{\K} d\A(\lambda)  \underset{w_{n+1} = w_0=0}{\int_{\mathscr{A}(0; \delta_N)}}\hspace{-.3cm} H^n( \lambda; \w) \prod_{j=0}^n K^\infty_{\eta_N(\lambda)}(w_{j}, w_{j+1})  d\A^{n}(\w)  \\
+ \notag\O\big( N L_N^{-2} \delta_N^3 \vee (\log N)^4 L_N^{-1} \big) .
\end{align}
By lemma~\ref{thm:delocalization}, the leading term in formula \eqref{cumulant_5} has the same limit (up to an arbitrary small error term) as
$$
\int_{\K} d\A(\lambda)  \underset{w_{n+1} = w_0=0}{\int_{\C^n}}\hspace{-.3cm} H^n( \lambda; \w) \prod_{j=0}^n K^\infty_{\eta_N(\lambda)}(w_{j}, w_{j+1})  d\A^{n}(\w) 
$$
and, since  $N L_N^{-2} \delta_N^3 \to0$ when $L_N \prec \sqrt{N}/(\log N)^3$, this completes the proof.  \qed\\


%Let $0<q_N<1$ be a sequence which converges to 0 as $N\to\infty$. Let $p_N = 1- q_N$ and denote $\widehat{K}^N = p_NK^N$.   


%By formula \eqref{cumulant}, proposition~\ref{thm:approximation} shows that the asymptotics of the cumulants of linear statistics of the  log-gases in dimension 2 is reduced to compute the integral of a multivariate polynomial of degree 2 against a product of Ginibre kernels.  
%This analysis has been performed in \cite{RV} and since our setting is a bit different, we will review the main steps in the rest of this section. \\

Since the function $\w\mapsto H^n(\lambda; \w )$ is a multivariate polynomial of degree 2,  the leading term in the asymptotics~\eqref{approximation} can be computed explicitly using the reproducing property of the \Ginibre kernel; see for instance~\cite{RV07b}. 
For any $\rho>0$, the function  $(z,w) \mapsto e^{\rho z \bar{w}}$ is the reproducing kernel for the Bergman space with weight  $\rho e^{-\rho|z|^2/2}$ on $\C$. This implies that for any $w_1, w_2 \in \C$ and for all integer $k\ge 0$,
\begin{equation*} %\label{reproducing}
\begin{cases}
\displaystyle \int_\C K^\infty_{\rho}(w_1,w_2) w_2^k K^\infty_{\rho}(w_2,w_3) d\A(w_2)  = w_1^kK^\infty_{\rho}(w_1,w_3)  \vspace{.2cm}   \\
%$$
%and taking complex conjugate:
%$$
\displaystyle \int_\C K^\infty_{\rho}(w_1,w_2) \bar{w_2}^k K^\infty_{\rho}(w_2,w_3) d\A(w_2)  = \bar{w_3}^kK^\infty_{\rho}(w_1,w_3)     
\end{cases} .
\end{equation*}

As a basic application  of these identities, we obtain the following lemma.





\begin{lemma} \label{thm:reproducing}
Let $n \ge 1$ and  $\rho>0$. For any polynomial $H(\w)$ of degree at most~2 in the variables $w_1,\dots, w_n, \bar{w_1}, \dots, \bar{w_n}$, we have
\begin{equation} \label{reproducing}
 \underset{w_{n+1} = w_0=0}{\int_{\C^n}}\hspace{-.3cm} H^n( \w) \prod_{j=0}^nK^\infty_{\rho}(w_{j}, w_{j+1}) \  d\A^n(\w)  
%& =  \int_{\C^n} \mho[H](w) \prod_{j=0}^nK^\infty_{\rho}(w_{j}, w_{j+1}) \  d\A^n(w)  \\
  = \sum_{1 \le r \le s\le n} \p_s \pb_r H |_{\w=0} . 
 \end{equation}
 %where 
 %$$ \mho[H](w) = \sum_{r, s =1}^n \p_r \pb_s H(0) w_r  \bar{w_s}  $$
%is the polynomial constructed from $H$ by keeping only the mixed terms.
\end{lemma}


According to formulae \eqref{normalization_0} and \eqref{normalization}, combining proposition~\ref{thm:approximation} and lemma~\ref{thm:reproducing},  we obtain for any test function $f\in \mathcal{C}^3_0(\K)$ and for all $n\in \N$ and $m\ge 0$,  
 \begin{align} \label{cumulant_6}
& \underset{z_0 = z_{n+1}}{\int_{\C^{n+1}}}\hspace{-.2cm}  \Upsilon^{n+1}_m[f_N](z_0, \z) \prod_{j=0}^n K^N_V(z_{j}, z_{j+1}) d\A(z_0) d\A^{n}(\z)  \\
&\notag
\hspace{1cm}= \sum_{2 \le r \le s \le n+1}\int_{\mathcal{K}}  \p_s \pb_r \Upsilon^{n+1}_m[f] (\lambda,\dots, \lambda)  d\A(\lambda)
+\O\left( \frac{(\log N)^4}{L_N} \right) .
\end{align}

In the macroscopic regime  ($L_N=1$ and $x_0=0$), 
this proves the estimate \eqref{assumption_1} as long as the support of the test function $\K\subset \S_V$, so that theorem~\ref{thm:crossover_2} follows directly from the asymptotic expansion \eqref{limit_1} as explained in section~\ref{sect:cumulant}.\\

In the mesoscopic regime, we claim  that the asymptotics \eqref{cumulant_6} with $m=0$ implies the central limit theorem~\ref{meso:2}.  The fact that the cumulants of order $n\ge 3$ vanish comes from the following combinatorial lemma.

\begin{lemma}[\cite{RV07b}, Lemma~9] \label{thm:RV}
 For any $n\ge 1$, let
\begin{align*}
\mathscr{Y}_n=
-\sum_{k\vdash n}
\Upsilon_0(\k) 
\left\{  \sum_{2 \le r < s\le \ell(\k)}  k_r k_s +   \sum_{r=2}^{\ell(\k)} k_r (k_r-n) \right\}.
  \end{align*}
  We have $\displaystyle
\mathscr{Y}_n = \begin{cases} 1 &\text{if } n=2 \\ 0 &\text{else} \end{cases} .
$
\end{lemma}
 
 
 
{\it Proof of theorem~\ref{meso:2}.} Let $\lambda\in\C$ and $\boldsymbol\lambda =(\lambda,\dots, \lambda)\in\C^{n+1}$. 
According to formula \eqref{Upsilon_f}, an elementary computation shows that for any $2 \le r < s\le n+1$,
\begin{equation} \label{combinatorics_1}
 \p_s \pb_r \Upsilon_0^{n+1}[f](\boldsymbol\lambda)
=   \p f(\lambda) \pb f(\lambda) f(\lambda)^{n-1} 
\sum_{\k \vdash n+1} \Upsilon_0(\k)  k_r k_s \1_{s \le \ell(k)}
 \end{equation}
 and
 \begin{align} \notag
 \p_r \pb_r \Upsilon_0^{n+1}[f]((\boldsymbol\lambda)
&=    \p f(\lambda) \pb f(\lambda) f(\lambda)^{n-1} 
\sum_{\k \vdash n+1} \Upsilon_0(\k)  k_r(k_r-1) \1_{r \le \ell(k)}\\
&\hspace{.2cm} \label{combinatorics_2}
+  \Delta f(\lambda)  f(\lambda)^{n}\sum_{\k \vdash n+1} \Upsilon_0(\k)  k_r \1_{r \le \ell(k)} .
 \end{align}
 
 
Since, by integration by parts, 
 $$
 \int_\C  \Delta f(\lambda)  f(\lambda)^{n} d\A(\lambda)
 = - n  \int_\C  \p f(\lambda) \pb f(\lambda) f(\lambda)^{n-1} d\A(\lambda) ,
 $$
we deduce from formulae \eqref{combinatorics_1} and \eqref{combinatorics_2} that
\begin{align*}
\sum_{2 \le r \le s \le n+1}\int_{\C}  \p_s \pb_r \Upsilon^{n+1}_m[f] (\boldsymbol\lambda)  d\A(\lambda) = -\mathscr{Y}_{n+1}
\int_{\C} \p f(\lambda) \pb f(\lambda) f(\lambda)^{n-1} d\A(\lambda) . 
  \end{align*}

When $ L_N =N^\alpha$ and  $0<\alpha<1/2$, formulae \eqref{cumulant_0} and \eqref{cumulant_6} with $m=0$  imply  that for any $n\ge 1$,
\begin{equation}\label{cumulant_7}
\lim_{N\to\infty}\Cu^{n+1}_{K^N_V}[f_N] = -\mathscr{Y}_{n+1}
\int_{\C} \p f(\lambda) \pb f(\lambda) f(\lambda)^{n-1} d\A(\lambda) . 
\end{equation}

By lemma~\ref{thm:RV}, this proves that for any test function $f\in\mathcal{C}^3_0(\C)$,  the centered mesoscopic linear statistics 
$\Xi(f_N) -\E\big[\Xi(f_N) \big] $ converges to a mean-zero Gaussian random variable $\mathrm{X}(f)$ with variance
$$ \Var\big[ \mathrm{X}(f) \big] =  \int_{\C} \p f(\lambda) \pb f(\lambda) d\A(\lambda) = \|f \|_{H^1(\C)}^2 ,  
$$
 as $N\to\infty$.  \qed\\


Using formula \eqref{cumulant} and the asymptotics~\eqref{cumulant_6}, we can  describe in a similar fashion the mesoscopic fluctuations of the incomplete log-gas $\hat{\Xi}$.  \\



{\it Proof of theorem~\ref{thm:crossover_3}.} 
 By lemma~\ref{thm:Bergman_kernel},  for any bounded function $f$ with compact support, we have
\begin{align*} 
 \int_\C f_N(z)^n  K^N_V(z,z) d\A(z)
 &= N  \int_\C f_N(z)^n 2 \Delta V(z) d\A(z) + \O(1) \\
 & = N L_N^{-2} \varrho_V(x_0)  \int_\C f(z)^n d\A(z) + \O(N L_N^{-3} ) .
\end{align*}
%where the equilibrium measure is $\varrho_V = 2 \Delta V \1_{\mathscr{D}_V}$. 
Let $T_N= N q_N  L_N^{-2} \varrho_V(x_0)$. By \eqref{cumulant} and \eqref{cumulant_6}, this shows that
    \begin{equation}  \label{cumulant_8}
\Cu^n_{\widehat{K}^N_V}[f]
= \Cu^n_{K^N_V}[f]  + T_N \gamma^n_m \int_\C f(z)^n d\A(z) + \underset{N\to\infty}{o}(T_N)
\end{equation}
as $N\to\infty$ and $q_N\to0$. As we already discussed in section~\ref{sect:cumulant}, the asymptotics 
\eqref{cumulant_7} and \eqref{cumulant_8} yield theorem~\ref{thm:crossover_3}. \qed\\



%%% Work -> the weighted potential theory problem 

\section{Transition for the 1-dimensional log-gases - the global regime. }\label{sect:1}


As we explained in the introduction,  there is an analogous transition for other determinantal processes and, in particular, for linear statistics of the eigenvalues of Hermitian random matrices or 1-dimensional log-gases.
A good reference for the results discussed in this section are the chapters 11--14 in the book of Pastur and Shcherbina, \cite{PS11}. 
Let $V: \R \to\R$ be a continuous function which  satisfies the condition~\eqref{potential} and let $\Xi$  and $\widehat{\Xi}$ be the determinantal processes with correlation kernels $K^N_V$ and $\widehat{K}^N_V = p_N K^N_V$ respectively. 
It was established in \cite{Johansson98} that  there exists an equilibrium measure $\mueq$ such that the one-point function
$u^N_V(x) : = N^{-1}K^N_V(x,x)$ viewed as a probability measure on $\R$ converges weakly to  $\mueq$ as $N\to\infty$.  
Like in the 2-dimensional setting, the equilibrium measure has compact support and it is absolutely continuous with  respect to  the Lebesgue measure with a density $\varrho_V$. Let 
\begin{equation} \label{bulk'}
\mathscr{I}_V = \{ x\in \R : \varrho_V(x)>0 \}  .
\end{equation}
We can always make an affine transformation of the potential  and assume that $\mathscr{I}_V \subset [-1,1] $. 
 Moreover, if $V$ is a polynomial and $\mathscr{I}_V =(-1,1)$,   then linear statistics of the  process $\Xi$ 
 satisfy a CLT:
\begin{equation} \label{Johansson}
\Xi(f) -  \int_\R f(x) \varrho_V(x) dx \Rightarrow \Y(f) \hspace{.5cm}\text{as }N\to\infty ,
\end{equation}
for any $f \in \mathcal{C}^2(\R)$ so that $f'(x)$ grows at most polynomially as $|x|\to\infty$. 
The process $\Y$ is a centered Gaussian noise defined on $[-1,1]$ with covariance structure:
\begin{equation} \label{Chebychev_covariance}
\E\big[ \Y(f)\Y(g) \big] = 
 \frac{1}{4}  \sum_{k=1}^\infty k \mathrm{c}_k(f)  \mathrm{c}_k(g) . 
 \end{equation}



Let $(T_k)_{k=0}^\infty$ denote the Chebyshev polynomials of the first kind, i.e.  $T_k(\cos \theta) = \cos(k\theta)$ for any $k\ge 0$. In formula \eqref{Chebychev_covariance}, $\mathrm{c}_k$ are the Fourier-Chebychev coefficients:
\begin{equation}\label{Fourier_T}
\mathrm{c}_k(f)=\frac{2}{\pi} \int_{-1}^1 f(x) T_k(x) \frac{dx}{\sqrt{1-x^2}} , 
\end{equation}
for any function $f\in L^\infty([-1,1])$.
The CLT \eqref{Johansson}  holds for more general potentials and for other  orthogonal polynomial ensembles as well, \cite[section~11.3]{PS11} or \cite{BG13a, BD16, Lambert_b} and   
it is  known that the one-cut condition, i.e.~the assumption that the support of the equilibrium measure is connected, is necessary. Otherwise, the asymptotic fluctuations of a generic linear statistic $\Xi(f)$ are still of  order~1 but there are not Gaussian, see \cite{Pastur06, Shcherbina13, BG13b}.  In this section,  we will establish the counterpart of the CLT \eqref{Johansson} for the  incomplete ensembles and for polynomial test functions. In particular, the estimate \eqref{assumption_1} will be proved using the approach  introduced in \cite{BD17} which consists in using the  three-terms recurrence relation of the orthogonal polynomials $\{P_k \}_{k=0}^\infty$ with respect to the measure 
$d\mu_N = e^{- 2N V(x)}dx$ to express the cumulants. 
For any $N\in \N$, there exists two sequences $a^N_k >0$ and $b^N_k \in \R$ such that the orthogonal polynomials $P_k$ satisfy 
\begin{equation}\label{recurrence}
xP_k(x) = a^N_k P_{k+1}(x) + b^N_k P_k(x) +  a^N_{k-1} P_{k-1}(x)  .
\end{equation}
%Under some mild conditions, the recurrence coefficients $(a^N_k, b^N_k)_{k=0}^\infty$ characterize the measure  $\mu_N$, see \cite[Chapter~2]{Deift99}, and their asymptotics comes in as a key condition in our main theorem. 

\begin{theorem} \label{thm:crossover_1}
Let $\mathrm{Y}$ be a mean-zero Gaussian process with covariance \eqref{Chebychev_covariance} and let  $\Lambda_{\tau\varrho_V}$ be an independent Poisson process with intensity $\tau\varrho_V$  where $\tau>0$. Let $p_N = 1-q_N$ and  $T_N = N q_N$. Suppose that the recurrence coefficients of the orthogonal polynomials $\{P_k \}_{k=0}^\infty$  satisfy for all $j\in\mathbb{Z}$, 
\begin{equation}\label{recurrence_limit} \lim_{N\to\infty} a^N_{N+j} = 1/2 \hspace{.5cm}\text{and}\hspace{.5cm}
   \lim_{N\to\infty}b^N_{N+j} = 0   ,
   \end{equation}
 and suppose that for all $k \in\N$, 
\begin{equation} \label{LLN_2}
\lim_{N\to\infty} \int_\R  x^k  u^N_V(x) dx  =   \int_\R x^k \varrho_V(x)  dx . 
\end{equation}
 Then, for any polynomial $Q\in\R\langle X\rangle$,  we obtain as $N\to\infty$ and $q_N \to 0$, 
\begin{align*}
%\label{Normal_1}
\hat{\Xi}(Q) - \E\big[\hat{\Xi}(Q) \big]  &\Rightarrow  \mathrm{Y}(Q) &\text{if }\ T_N \to 0 , \hspace{.2cm}\\ 
%\label{Qoisson_1}
\frac{\hat{\Xi}(Q) - \E\big[\hat{\Xi}(Q) \big]}{\sqrt{T_N} } &  \Rightarrow
 \No\left(0, \int_{\R}  Q(x)^2 \varrho_V(x) dx \right)   
&\text{if }\ T_N \to \infty ,\\
%\label{crossover_1}
\hat{\Xi}(Q) - \E\big[\hat{\Xi}(Q) \big] & \Rightarrow  \mathrm{Y}(Q) + \Lambda_{\tau\varrho_V}(-Q) &\text{if }\ T_N \to \tau  . \hspace{.25cm}
\end{align*}
\end{theorem}


The condition \eqref{LLN_2} can easily be deduced from the following estimate for the one-point function or density of the gas,
$$
u^N_V(x) = N^{-1}K^N_V(x,x) \le e^{- 2N \{ V(x) - \log(1+|x|) - C\} } ,
$$
where $C$ is a constant which depends only on the potential $V$; c.f.~ \cite[Theorem~11.1.2]{PS11}. %p331
 In particular, the assumption \eqref{potential} implies that the function $u_N(x)$ decays faster than any polynomial outside a compact set and  
the limits \eqref{LLN_2} can be deduced from the weak convergence of $u_N (x)$ to the equilibrium measure $d\mueq = \varrho_V dx$. It was proved in~\cite{Johansson98} that if $V$ is a convex polynomial and $\mathscr{I}_V =(-1,1)$, then the condition \eqref{recurrence_limit} is satisfied and theorem~\ref{thm:crossover_1} holds. In fact, this assumption should be generic and it is an interesting question whether it is equivalent to the condition $\mathscr{I}_V =(-1,1)$. \\

% Moreover, it was proved in \cite{Lambert_b} that if there exists two continuous functions  $a: \R_+ \to\R_+ $ and $b:\R_+ \to\R$ 
%such that the recurrence coefficient satisfies for all  $t \in [0,1]$ and for all $k\in\mathbb{Z}$, 
%\begin{equation*}
 %\lim_{N\to\infty} a^N_{\lfloor Nt  \rfloor+k}= a(t) \hspace{.5cm}\text{and}\hspace{.5cm}
%\lim_{N\to\infty }b^N_{\lfloor Nt  \rfloor+k}  = b(t)  , 
%\end{equation*}
%then

 


The completion $\mathscr{L}_N$ of the space of polynomials with respect to $L^2(\R,\mu_N)$ is isomorphic to $L^2(\N_0)$ and formula \eqref{recurrence} implies that the multiplication by $x$ on $\mathscr{L}_N$ is unitary equivalent to the matrix
\begin{equation}\label{Jacobi}
\mathbf{J} := \begin{bmatrix}
 b_0^N & a_0^N & 0 & 0 & 0  \\
 a_0^N & b_1^N & a_1^N & 0 & 0  &\mathbf{0} \\
  0 &  a_1^N  &  b_2^N  &  a_2^N  & 0 \\ 
 0 & 0 &  a_2^N &  b_3^N &  a_3^N \\
 &\mathbf{0} &  & \ddots &\ddots&\ddots
\end{bmatrix} .
\end{equation}   
This matrix is called the {\bf Jacobi matrix}. 
%
%
%If $\J_n$ denotes the $n\times n$ principal minor of the 
%Jacobi matrix,  then $P_n(x) = \gamma_n \det(x-\J_n) $ 
%
%
The connection with linear statistics of eigenvalues comes from the fact that for any polynomial
$Q\in\R\langle X\rangle$ and  for any composition $\k\vdash n$, one easily sees that
\begin{align*} \notag
&\underset{x_{0} =x_l}{\int_{\X^l}} Q(x_1)^{k_1} \cdots  Q(x_l)^{k_l} \prod_{0\le j\le l} K^N_V(x_j, x_{j-1}) dx_1 \cdots dx_l \\
& %\label{cumulant_9} 
\hspace{2cm}= \sum_{m=0}^{N-1} \sum_{\pi \in \Gamma_{m}^n }   \prod_{j=1}^l \1_{\pi(k_1+\cdots+ k_j) < N} \prod_{i=0}^{n-1}
Q(\J)_{\pi(i)\pi(i+1)}.
\end{align*}
where $\mathcal{G}$ denotes the   adjacency graph of the matrix $Q(\J)$ and
$$
\Gamma_{m}^n = \big\{ \text{paths }\pi  \text{ on the graph }\mathcal{G}\text{ of length }n \text{ such that }\pi(0)=\pi(n)=m  \big\} .
$$
For any composition $\k\vdash n$, we let
\begin{equation}
 \Phi_\pi^N(\k) := \1_{\hspace{-.1cm}\displaystyle\max_{1\le j <\ell(\k)}\pi(k_1+\cdots+ k_j) \ge N } .
\end{equation}
Observe that
$$
\prod_{1\le j <\ell(\k)} \1_{\pi(k_1+\cdots+ k_j) < N}  = 1-  \Phi_\pi^N(\k) , 
$$
so that by formula \eqref{cumulant_0},  the cumulants of a polynomial linear statistics are given by 
\begin{equation} \label{cumulant_15}
\Cu^n_{K^N_V}[Q]
= - \sum_{m =0}^{N-1} \sum_{\pi \in \Gamma_{m}^n } \prod_{i=0}^{n-1}
Q(\J)_{\pi(i)\pi(i+1)}  \sum_{\k \vdash n}\Upsilon_0(\k)\big( 1-\Phi_\pi^N(\k)\big) . 
\end{equation}

Note that there exists a constant $M>0$ which only depends on the degree of $Q$ and $n$ so that  $\Phi_\pi^N=0 $ for any path $\pi \in \Gamma^n_m$ as long as $m < N - M$. 
Since $  \sum_{\k \vdash n}\Upsilon_0(\k) = 0$ for all $n \ge 2$, formula \eqref{cumulant_15} implies that  
\begin{equation*}
\Cu^n_{K^N_V}[Q]
=  \sum_{m =N -M}^{N-1} \sum_{\pi \in \Gamma_{m}^n } \prod_{i=0}^{n-1}
Q(\J)_{\pi(i)\pi(i+1)}  \sum_{\k \vdash n}\Upsilon_0(\k)\Phi_\pi^N(\k)  . 
\end{equation*}

In particular, if the matrix $\J$ has a right-limit, i.e.~there exists an (infinite) matrix $\L$ such that for all $i,j \in\mathbb{Z}$, 
\begin{equation}\label{right_limit}
\lim_{N\to\infty} \J_{N+i, N+j} = \L_{i,j} , 
\end{equation}
then 
\begin{equation}
\lim_{N\to\infty}\Cu^n_{K^N_V}[Q]
=  \sum_{m =1}^{M} \sum_{\pi \in \widetilde{\Gamma}_{m}^n } \prod_{i=0}^{n-1}
Q(\L)_{\pi(i)\pi(i+1)}  \sum_{\k \vdash n}\Upsilon_0(\k)\Phi_\pi^0(\k) , 
\end{equation}
where $\widetilde{\mathcal{G}}$ denotes the   adjacency graph of the matrix $Q(\L)$ and
$$
\widetilde{\Gamma}_{m}^n = \big\{ \text{paths }\pi  \text{ on the graph }\widetilde{\mathcal{G}}\text{ of length }n \text{ such that }\pi(0)=\pi(n)=-m  \big\} . 
$$

Since  $\sum_{\k \vdash n} \Upsilon_m(\k) = 0$ for all $n, m\in\N$, 
the very same computation shows that
\begin{align} \notag
&\lim_{N\to\infty} \underset{x_{0} =x_n}{\int_{\X^n}}     \Upsilon^n_m[Q](x)  \prod_{1\le j\le n} K^N_V(x_j, x_{j-1}) d^nx \\
&\label{cumulant_16}\hspace{2cm}
=  \sum_{m =1}^{M} \sum_{\pi \in \widetilde{\Gamma}_{m}^n } \prod_{i=0}^{n-1}
Q(\L)_{\pi(i)\pi(i+1)}  \sum_{\k \vdash n}\Upsilon_m(\k)\Phi_\pi^0(\k) . 
\end{align}

Hence, the condition \eqref{assumption_1} obviously holds  and, by formula \eqref{LLN_2}, this yields that  for all $n \ge 2$, 
\begin{equation*}
\Cu^n_{\widehat{K}^N_V}[Q]=\Cu^n_{K^N_V}[Q] + Nq_N \int_\R  Q(x)^{n}  \varrho_V(z) dx  
+ \O(Nq_N^2)  .
 \end{equation*}
The condition \eqref{recurrence_limit} implies that the right-limit of the Jacobi matrix is a tridiagonal matrix $\L$ such that $\L_{jj}= 0$ and $\L_{j, j\pm 1} =1/2$   for all $j \in \mathbb{Z}$ and it  was proved in \cite{Lambert_b}, see also~\cite{BD17}, that in this case:
$$
\lim_{N\to\infty} \Cu^n_{K^N_V}[Q]  = \begin{cases}
\displaystyle 
 \frac{1}{\pi^2}  \sum_{k=1}^{\operatorname{deg} Q}  k \bigg( \int_{-1}^1 Q(x) T_k(x) \frac{dx}{\sqrt{1-x^2}}  \bigg)^2  &\text{if } n=2 \\
 0 &\text{else} 
\end{cases} .
$$
As we discussed in section~\ref{sect:cumulant}, these asymptotics imply
theorem~\ref{thm:crossover_1}. 


\begin{remark}[Generalizations of theorem~\ref{thm:crossover_1}] \label{rk:biorthogonal}
Note that we have formulated theorem~\ref{thm:crossover_1} for a log-gas at inverse temperature $\beta=2$, but the previous proof can be  generalized to other biorthogonal ensemble with a correlation kernel of the form
  \begin{equation*} 
K^N(z,w) = \sum_{k=0}^{N-1} \varphi_k^N(z) \varpi_k^N(w) .  
\end{equation*} 
The appropriate assumptions are that the equilibrium measure $\mueq$ exists, formula \eqref{LLN_1} holds for all polynomials,  the family $\{\varphi_k^N \}_{k=0}^\infty $ satisfies a $q$-term recurrence relation for all $N\in\N$  and the corresponding recurrence matrix $\J$  has a right-limit~$\L$ as $N\to\infty$.  
 This applies to other orthogonal polynomial ensembles, such as the discrete point processes coming from domino tilings of hexagons, as well as some  non-symmetric biorthogonal  ensembles  such as the Muttalib-Borodin ensembles, square singular values of product of complex Ginibre matrices or certain two-matrix models,  see \cite{BD17, Lambert_b} for more details.
Note also that  formula    \eqref{cumulant_16} still  holds in the case where the right-limit $\L$ exists but is not Toeplitz (i.e.~constant along its main diagonals). Then, we obtain a crossover from a non-Gaussian process (described by $\L$ in the regime where $T_N\to 0$) to a Poisson process (when $T_N\to \infty$). 
For instance, such a transition arises when considering linear statistics of the log-gases in the multi-cut regime.    
 \end{remark}


\section{Transition for the 1-dimensional log-gases - the mesoscopic regime. }\label{sect:1'}

The combinatorial method used in the previous section is well-suited to investigate the global properties of the processes $\Xi$ and $\hat{\Xi}$ but it is difficult to implement in the mesoscopic regime (in this case, polynomial linear statistics do not converge without normalization). Although there is a mesoscopic theory based on the asymptotics of the recurrence matrix \eqref{Jacobi}
 developed in \cite{BD16}, it is simpler to prove
%
%, although it has been applied successfully to a certain class of orthogonal polynomial ensembles in \cite{BD16}
 the mesoscopic counterpart of theorem~\ref{thm:crossover_1} using the sine-kernel 
asymptotics of the correlation kernel $K^N_V$ and the method of~\cite{Lambert_a}. This reduces the proof  to carefully apply the classical asymptotics  of \cite{Deift99+}. These results of Deift et al.~rely on the Riemann-Hilbert problem associated to the orthogonal polynomials $\{P_k \}_{k=0}^\infty$ and holds as long as the potential $V$ is real-analytic (including the multi-cut and non-regular cases). The steepest descent method of \cite{Deift99+} has also been established for many other invariant ensembles, c.f.~\cite{Kuijlaars11}, for which it is also possible to extract the asymptotics  \eqref{sine_1} for the correlation and deduce the analogue of theorem~\ref{thm:crossover_4}.
  We let $x_0\in\mathscr{I}_V$, $f\in \mathcal{C}^2_0(\R)$, $\alpha\in (0,1)$,  and $f_N(x) = f\big(N^\alpha(x-x_0) \big)$. In the following, we denote by
$$
\hat{f}(u) = \int_\R f(x) e^{-2\pi i x u} dx
$$
the Fourier transform of the function $f$. 

%
%
% the asymptotics of the correlation kernel $K^N_V$ can be derived at all mesoscopic scales using the Riemann-Hilbert formulation of \cite{} and the steepest descent of \cite{}. 
%


\begin{theorem} \label{thm:crossover_4}

Let $\Z$ be a mean-zero Gaussian process on $\R$ with correlation structure:
\begin{equation}
\E\big[ \Z(h)\Z(g) \big] = \int_{0}^\infty u \hat{h}(u) \hat{g}(-u) du
\end{equation} 
for any functions $h, g \in H^{1/2}(\R)$  and let $\Lambda_{\tau}$ be an independent  Poisson process with intensity $\tau>0$ on $\R$. Let $p_N = 1-q_N$ and  $T_N =  q_NN^{1-\alpha} \varrho_V(x_0)$.\\
 We obtain  as $N\to\infty$ and $q_N\to0$, 
\begin{align}
\label{Normal_4}
\hat{\Xi}(f_N) - \E\big[\hat{\Xi}(f_N) \big]  &\Rightarrow \Z(f) &\text{if }\ T_N \to 0 , \hspace{.2cm}\\ 
\label{f_Noisson_4}
\frac{\hat{\Xi}(f_N) - \E\big[\hat{\Xi}(f_N) \big]}{\sqrt{T_N} } &  \Rightarrow
 \No\left(0, \int_{\R}  f(x)^2 dx \right)   
&\text{if }\ T_N \to \infty ,\\
\label{crossover_4}
\hat{\Xi}(f_N) - \E\big[\hat{\Xi}(f_N) \big] & \Rightarrow \Z(f) + \Lambda_{\tau}(-f) &\text{if }\ T_N \to \tau  . \hspace{.25cm}
\end{align}
\end{theorem}

Since $\Var\big[ \Z(f)\big] = \| f\|_{H^{1/2}(\R)}$, the process $\Z$ is usually called the $H^{1/2}$-Gaussian noise. It describes the mesoscopic fluctuations of the eigenvalues of Hermitian random matrices, see \cite{BD16, Lambert_a,  LS15, HK16}, as well as  the mesoscopic fluctuations of the log-gases for general $\beta>0$, \cite{BL16}, and of random band matrices in the appropriate regime \cite{EK14a, EK14b}. \\
% As we mentioned in  the introduction, analogous transitions with other mechanisms have already  been investigated for the GUE eigenvalue process  and the crossovers are different. \\


%In contrast to formula \eqref{crossover_4},  it was proved in~\cite{DJ16} that for Dyson's Brownian motion the critical fluctuations are Gaussian and for the  {\it positive temperature GUE} investigated in \cite{JL15} the critical fluctuations are also not Gaussian but they are described  by complicated formulae.  



Our proof of theorem~\ref{thm:crossover_4}  is based on the strategy explained in section~\ref{sect:cumulant} and 
%we rely on the sine-kernel asymptotics of the correlation kernel $K^N_V$, see formula \eqref{sine_1} below, to derive the necessary  cumulants' estimate.
 is postponed to the end of this section after we review some of the ideas of~\cite{Lambert_a}. 
 By making a  change of variables in formula \eqref{cumulant},  the cumulants of the linear statistics of the incomplete ensembles are given by
    \begin{align}  \label{cumulant_9}
\Cu^n_{\widehat{K}^N_V}[f_N]
= \Cu^n_{K^N_V}[f_N]  
&- \sum_{m=1}^n (-q_N)^m  \gamma^n_m \int_\R  f(x)^{n} \widetilde{K}^N_{V,x_0}(x,x) dx  \\
&\notag-    \sum_{m=1}^n (-q_N)^m  \underset{x_{0} =x_n}{\int_{\R^n}}     \Upsilon^n_m[f](x)  \prod_{j\le n} \widetilde{K}^N_{V,x_0}(x_j, x_{j-1}) dx  ,
\end{align}
where $\widetilde{K}^N_{V,x_0}$ denotes the rescaled correlation kernel
$$\widetilde{K}^N_{V,x_0}(x, y) = \frac{1}{N^\alpha} K_{V}^N\left(x_0+\frac{x}{N^{\alpha}} ,x_0 +\frac{y}{N^{\alpha}}\right)   . 
$$ 
Recall that $\varrho_V$ is the density of  the equilibrium measure and define {\it the integrated density of states}:
\begin{equation} \label{IDS}
F_V(x) = \int_0^x  \varrho_V(s) ds . 
 \end{equation}


Using the recurrence relation \eqref{recurrence}, one can rewrite the correlation kernel as
  \begin{equation*}
K^N_V(x,y) = a^N_{N-1}  \frac{\varphi_N(x) \varphi_{N-1}(y) - \varphi_N(y) \varphi_{N-1}(x)}{x-y} . 
\end{equation*}



Then,  based on the results of \cite{Deift99+},  one  can show that for any $\alpha \in (0,1]$, we have
\begin{equation} \label{sine_1}
\widetilde{K}^N_{V,x_0}(x, y)   
= \frac{\sin \left[ \pi  N \big(( F_V(x_0+ x N^{-\alpha})-F_V(x_0+ y N^{-\alpha})\big)\right]}{\pi(x-y)}+ \underset{N\to\infty}{O}\left(N^{-\alpha}\right)  ,
\end{equation}
uniformly for all $x_0$ in compact subsets of $\mathscr{I}_V$ and for all $x, y$ in compact subsets of $\R$; c.f.~\cite[Poposition~3.5]{Lambert_a}. 
In particular,  by continuity along the diagonal, for any $x_0\in \mathscr{I}_V$, 
$$\widetilde{K}^N_{V,x_0}(x, x)  = N^{1-\alpha} \varrho_V(x_0)  + \underset{N\to\infty}{O}\left(N^{-\alpha}\right) ,   $$
and this implies that for all $n\in\N$, 
\begin{equation} \label{LLN_4}
\int_\R  f(x)^{n} \widetilde{K}^N_{V,x_0}(x,x) dx
\simeq N^{1-\alpha} \varrho_V(x_0)  \int_\R  f(x)^{n} dx .
\end{equation}






We define the sine kernel with density $\rho>0$ on $\R$ by
\begin{equation} \label{sine}
K^{\sin}_\rho(x,y) = \frac{\sin[\pi \rho(x-y)]}{\pi(x-y)} .
\end{equation}
We see by taking $\alpha=1$ in formula \eqref{sine_1} that  the sine process with correlation kernel \eqref{sine} describe the local limit in the bulk of the 1-dimensional log-gases. 
In the mesoscopic regime, it was proved in \cite{Lambert_a} that, up to a change of variable, it is possible to replace the kernel $\widetilde{K}^N_{V,x_0}$ by an appropriate sine-kernel using the asymptotics \eqref{sine_1} in the cumulant formulae. Namely, for any $n\ge 2$, 
\begin{equation} \label{sine_2}
\Cu^n_{K^N_V}[f_N] \simeq  \Cu^n_{K^{\sin}_{\eta_N(x_0) }}[ f\circ\zeta_N]
\end{equation}
where  
\begin{equation} \label{zeta_N}
\zeta_N(x) =  N^\alpha \left\{ G_V\left( F_V(x_0) + \varrho_V(x_0) \frac{x}{N^\alpha} \right)-x_0\right\}  . 
\end{equation}
Here, the function $G_V$ denotes the inverse of the integrated density of sates $F_V$, \eqref{IDS}. \
By the inverse function theorem, it exists in a neighborhood of any point $F_V(x_0)$ when $x_0\in\mathscr{I}_V$ and the map  $\zeta_N$ is well-defined  on any compact subset of $\R$ as long as the  parameter $N$ is sufficiently large.
The combinatorics of the cumulants of linear statistics of a determinantal process are associated with the map  $\Upsilon_0$, see formulae \eqref{cumulant_0}--\eqref{Upsilon_0}. 
However, we deduce from the proof of proposition~2.2 in \cite{Lambert_a} that a similar asymptotics holds for  any function $\Upsilon:\mho\to \R$. In   particular, we get  for any $m \ge 1$, 
\begin{equation}\label{cumulant_10}
 \underset{x_{0} =x_n}{\int_{\R^n}}     \Upsilon^n_m[f](x)  \prod_{j\le n} \widetilde{K}^N_{V,x_0}(x_j, x_{j-1}) d^nx \
 \simeq \underset{x_{0} =x_n}{\int_{\X^n}}  \hspace{-.2cm}   \Upsilon^n_m[h_N](x)  \prod_{1\le j\le n} K^{\sin}_{\eta_N}(x_j, x_{j-1}) d^nx
\end{equation}
where $h_N = f\circ \zeta_N$. 




\begin{proposition} \label{thm:g}
%Let  $x_0\in \mathscr{I}_V$, $0<\alpha<1$, $f \in C^2_0(\R)$.
Suppose that $\supp(f) \subset (-L, L)$.  There exists 
$N_0 >0$ such that for all $N \ge N_0$,
 the functions $h_N= f\circ \zeta_N$ are well-defined on $\R$,  $ h_N\in  C^2_0([-L,L])$ and for all $u\in\R$, 
\begin{equation} \label{g_estimate} 
\big| \widehat{h_N}(u) \big| \le \| f \|_{\mathcal{C}^2(\R)} \frac{C }{1+ |u|^2} .
\end{equation} 
\end{proposition}

\proof When the potential $V$ is analytic, the {\it bulk} $\mathscr{I}_V$ is composed on finitely many open intervals and the equilibrium density $\varrho_V$ is smooth on $\mathscr{I}_V$. Since $x_0\in\mathscr{I}_V$, by formula \eqref{zeta_N}, the function $\zeta_N$ is increasing and smooth on the interval $[-L,L]$, and we have
$$  \zeta_N''(x) =   \varrho_V(x_0)^2 G_V''\big(F_V(x_0) + \varrho_V(x_0) xN^{-\alpha}\big ) N^{-\alpha}. 
$$
Moreover, since $\zeta_N(0)=0$ and  $\zeta'_N(0) =G_V'\big(F_V(x_0)\big) \varrho_V(x_0) =1$, this implies that 
$$  \zeta_N(x) = x + \O(N^{-\alpha}) $$  
uniformly for all $x\in[-L,L]$.
Since the interval $(-L, L)$ contains the support of the test function $f$, this estimate shows that when the parameter $N$ is large, we can define  $h_N(x) = f\big( \zeta_N(x) \big)$ for all $x\in[-L, L]$ and extend it by 0 on $\R \backslash [-L,L]$.  
Then  $h_N \in  C^2_0(\R)$ and 
\begin{equation} \label{h_N''}
h_N''(x) =  \zeta_N''(x) f'(\zeta_N(x)) +  \zeta_N'(x)^2f''(\zeta_N(x)) 
\end{equation}
for all $x\in[-L,L]$. Moreover, we can use the estimate
\begin{equation} \label{Fourier_estimate} 
\big| \widehat{h_N}(u) \big| \le C \frac{\| h_N \|_\infty + \| h_N''\|_\infty}{1+ |u|^2} 
\end{equation}
to get the upper-bound \eqref{g_estimate}. Plainly $\| h_N \|_\infty  \le \| f\|_\infty$ and it is easy to  deduce from formula \eqref{h_N''} that
$$ \| h_N'' \|_\infty \le \| f' \|_\infty + C \|f''\|_\infty . $$  
\qed




For any $u \in \R^n$ and for any composition $\k\vdash n$, define 
$$ \Psi_u(\k) =  2\max_{1\le j <\ell(\k)}\{0, u_1+\cdots + u_{k_1+\cdots +k_j} \} .
$$

To compute the limit of the RHS of \eqref{cumulant_10}, we need the following asymptotics. 

\begin{lemma} \label{thm:sine}
Let $n\ge 2$ and let $\eta_N>0$ such that $\eta_N\nearrow \infty$.  Suppose that $h_N$ is a sequence of integrable functions such that 
\begin{equation}\label{assumption_2}
\lim_{N\to\infty} \hspace{-.3cm}
\underset{\begin{subarray}{c} u_1+\cdots +u_{n}=0  \\ |u_1|+\cdots +|u_n| > \eta_N \end{subarray}}{\int_{\R^{n-1}}} \hspace{-.4cm} \big| \widehat{h_N}(u_1)\cdots   \widehat{h_N}(u_n) \big| |u_1|  d^{n-1}\mathrm{u} =0 . 
\end{equation}
Then,  for any map $\Upsilon: \mho \to\R$ such that $\sum_{\k \vdash n} \Upsilon(\k) =0$, we have
\begin{equation}\label{cumulant_11}
 \hspace{-.5cm}
\underset{x_{0} =x_n}{\int_{\X^n}}  \hspace{-.2cm}   \Upsilon^n[h_N](x)  \prod_{1\le j\le n} K^{\sin}_{\eta_N}(x_j, x_{j-1}) d^nx\
\simeq -
 \hspace{-.3cm}\underset{u_1+\cdots +u_n=0}{\int_{\R^{n-1}}} \hspace{-.3cm}  \Re\bigg\{ \prod_{j=1}^n  \widehat{h_N}(u_j)  \bigg\}\sum_{\k \vdash n}\Upsilon(\k)\Psi_u(\k)  d^{n-1}u .
\end{equation}
\end{lemma}

\proof The argument is the same as the proof of Lemma~1 in Soshnikov's paper \cite{Soshnikov00} on linear statistics of the CUE, see also \cite{Spohn87} or  section~2.1 in \cite{JL15}. 
Based on the formula
$$
K^{\sin}_{\eta_N}(x,y) = \int_\R \1_{\{ |u|< \eta_N/2\} } e^{2\pi i u (x-y)}  du , 
$$
we obtain 
\begin{align*}
\mathscr{T}_N&:=\underset{x_{0} =x_n}{\int_{\X^n}}  \hspace{-.2cm}   \Upsilon^n[h_N](x)  \prod_{1\le j\le n} K^{\sin}_{\eta_N}(x_j, x_{j-1}) d^nx \\
& = 
 \hspace{-.2cm}\underset{u_1+\cdots +u_n=0 \hspace{.1cm}}{\int_{\R^{n-1}}}  \prod_{j=1}^n  \widehat{h_N}(u_j)  \sum_{\k \vdash n}\Upsilon(\k) \max\big\{0, \eta_N - \Psi_u(\k)/2 - \Psi_{-u}(\k)/2 \big\} d^{n-1}u .
\end{align*}
Then, the condition $\sum_{\k \vdash n} \Upsilon(\k) =0$ implies that 
\begin{align*}
&\bigg| \mathscr{T}_N +
 \hspace{-.2cm}\underset{u_1+\cdots +u_n=0 \hspace{.1cm}}{\int_{\R^{n-1}}}  \prod_{j=1}^n  \widehat{h_N}(u_j)  \sum_{\k \vdash n}\Upsilon(\k) \frac{ \Psi_u(\k) + \Psi_{-u}(\k)}{2} d^{n-1}u \bigg| \\
 & \le 
 \hspace{-.2cm}\underset{u_1+\cdots +u_n=0 \hspace{.1cm}}{\int_{\R^{n-1}}}
   \bigg| \prod_{j=1}^n  \widehat{h_N}(u_j) \bigg|  \sum_{\k \vdash n} |\Upsilon(\k) |  \Psi_u(\k)    \1_{ \{\Psi_u(\k) + \Psi_{-u}(\k) \ge 2 \eta_N \} }d^{n-1}u 
\end{align*}

Since $ \big| \Psi_u(\k)/2 \big| \le |u_1|+\cdots +|u_n|$ for any $\k\vdash n$, the condition \eqref{assumption_2} is sufficient to obtain formula \eqref{cumulant_11}. \qed\\



{\it Proof of theorem~\ref{thm:crossover_4}.}
Using the estimate \eqref{g_estimate}, we see that when the parameter is sufficiently large,  there exists a constant $C>0$ so that  
\begin{align} \notag 
&\Bigg| \underset{u_1+\cdots +u_n=0}{\int_{\R^{n-1}}} \hspace{-.3cm}  \Re\bigg\{ \prod_{j=1}^n  \widehat{h_N}(u_j)  \bigg\}\sum_{\k \vdash n}\Upsilon_m(\k)\Psi_u(\k)  d^{n-1}u  \Bigg| \\
&\label{cumulant_12} \hspace{2cm}
\le C \hspace{-.2cm}  \underset{u_1+\cdots +u_n=0}{\int_{\R^{n-1}}}  \frac{|u_1|+\cdots +|u_n|}{(1+ |u_1|^2) \cdots (1+|u_n|^2)} d^{n-1}u ,
\end{align} 
A similar upper-bound shows that the sequence $h_N = f\circ \zeta_N$ satisfies the condition \eqref{assumption_2} of lemma~\ref{thm:sine}. Hence, combining the asymptotics \eqref{cumulant_10} and \eqref{cumulant_11}, we obtain
\begin{align}\label{cumulant_13}
& \underset{x_{0} =x_n}{\int_{\R^n}}     \Upsilon^n_m[f](x)  \prod_{j\le n} \widetilde{K}^N_{V,x_0}(x_j, x_{j-1}) d^nx \\
 &\notag \hspace{2cm}\simeq -
 \hspace{-.3cm}\underset{u_1+\cdots +u_n=0}{\int_{\R^{n-1}}} \hspace{-.3cm}  \Re\bigg\{ \prod_{j=1}^n  \widehat{h_N}(u_j)  \bigg\}\sum_{\k \vdash n}\Upsilon_m(\k)\Psi_u(\k)  d^{n-1}u .
\end{align}
Thus,  by \eqref{cumulant_12}, the integral~\eqref{cumulant_13} is uniformly bounded in $N$ for any $m \ge 1$. This property combined with the asymptotics \eqref{LLN_4} and  formula \eqref{cumulant_9} implies that for all $n \ge 2$, 
\begin{equation} \label{cumulant_14}
\Cu^n_{\widehat{K}^N_V}[f_N]
= \Cu^n_{K^N_V}[f_N]  
+ \gamma_1^n  q_NN^{1-\alpha} \varrho_V(x_0)  \int_\R  f(x)^{n} dx + \O(q_N^2N^{1-\alpha} ) .
\end{equation}

Then, using the asymptotics \eqref{sine_2} and Soshnikov's main combinatorial lemma, it was proved in \cite{Lambert_a} that
$$
\lim_{N\to\infty} \Cu^n_{K^N_V}[f_N]   = \begin{cases}  \displaystyle \int_{0}^\infty u \big| \hat{f}(u)  \big|^2 du &\text{if } n=2  \\ 
0  &\text{if } n \ge 3 \end{cases}.
$$
Hence, the expansion \eqref{cumulant_14} yields the weak convergence results \eqref{Normal_4}--\eqref{crossover_4} in the different regimes. \qed


\begin{appendices}

\appendix\section{Incomplete determinantal processes} \label{A:kernel}

In this appendix, we review some background material on determinantal processes and we give an alternative proof of the results of Bohigas and Pato, \cite{BP06}, that the incomplete process is determinantal with correlation kernel $p_NK^N_V(z,w)$. 
There are many excellent surveys about  determinantal processes and we refer to 
\cite{Soshnikov, Konig, Johansson06, Borodin11} for further details and an overview of the  main examples. 



Let $\X$ be a complete separable metric space equipped with a Radon measure~$\mu$. 
The configuration space $\mathscr{Q}(\X)$ is the set of integer-valued locally finite Borel measure on $\X$ equipped with the topology which is generated by the maps 
$$\mathscr{Q}(\X) \ni \Xi \mapsto \Xi_A := \int_A d\Xi    $$ for every Borel set $A\subseteq \X$.
A point process $\mathbb{P}$ is a Borel probability measure on  the configuration  space  $\mathscr{Q}(\X)$. 
A point process can be characterized by its correlation functions $(\rho_n)_{n=1}^\infty$. If it exists, $\rho_n$ is a symmetric function on $\X^n$ which satisfies the identity   
\begin{equation}\label{correlation}
\E\left[  {\Xi_{A_1} \choose k_1} \cdots   {\Xi_{A_\ell} \choose k_\ell} \right] = \frac{1}{k_1!\cdots k_\ell!} \int_{A_1^{k_1} \times \cdots \times A_\ell^{k_\ell}} \rho_n(x_1,\dots , x_n) d\mu^n(x)
\end{equation}
 for all composition $\k \vdash n$  and for all disjoint Borel sets $A_1,\dots, A_\ell \subseteq \X$. 
A point process is called {\bf determinantal} if all its correlation functions exist are given by
\begin{equation} \label{det}
\rho_n(x_1,\dots , x_n) = \det_{n\times n}\big[ K(x_i, x_j) \big] . 
\end{equation}
The function $K: \X \times \X \to\C$ is called the {\bf correlation kernel}. 
For instance,  given random points $(\lambda_1,\dots, \lambda_N)$ with the joint density $G_N(x)=e^{-\beta \mathscr{H}^N_V(x)}/Z^N_V$ on $\X^N$, see \eqref{log_gas}, the random measure $ \sum_{k=1}^N \delta_{\lambda_k}$ defines a point process  and its correlation functions  satisfy $\rho_N = N! G_N$ and 
\begin{equation}\label{correlation_function}
\rho_k(x_1, \dots, x_k) = \frac{N!}{(N-k)!}   \int  G_N(x_1, \dots, x_N) dx_{k+1}\cdots dx_N
\end{equation}
for all $k<N$. When $\beta=2$, it is easy to verify that 
$$ G_N(x) = \frac{1}{N!} \det_{N\times N} \big[K^N_V(x_i, x_j) \big]$$ 
and that, by formula \eqref{correlation_function},  the process $\Xi$ is determinantal with the correlation kernel 
$K^N_V$ given by \eqref{kernel}. In general, if  $K\in L^2(\X\times \X, \mu\times \mu)$  is a symmetric function which satisfies the property \eqref{projection}, then there exists a determinantal process on $\X$ with correlation kernel $K$; c.f.~\cite[Theorem~3]{Soshnikov}. \\





%In this general setting, let $\Xi and  

Let $q\in (0,1)$ and $p=1-q$.
Recall that $X$ is a Binomial random variable with parameter $p$ and $N\in\N$ if
$$
\P[X = n] = {N \choose n} p^n q^{N-n}
$$ 
for all $n=0,\dots , N$. Moreover, we can check that for any $k \in \N_0$,  
\begin{equation} \label{binomial_moment}
 \E\left[{X \choose k} \right]   = p^k {N \choose k} . 
 \end{equation}

Let $\Xi$ be a random (point) configuration and let $\widehat{\Xi}$ be the configuration obtained after performing a Bernoulli percolation on  $\Xi$. 
By construction,  for any disjoint Borel set $A\subseteq \X $, the conditional distribution of   the random variable $\widehat{\Xi}_A$ given $\Xi$  has a Binomial distribution with parameters $p$ and $\Xi_A$ and it is statistically independent of $\Xi_B$ for any Borel set $B$ disjoint of $A$. 
By formula \eqref{binomial_moment}, this implies that
\begin{align*}
\E\left[  {\widehat\Xi_{A_1} \choose k_1} \cdots   {\widehat\Xi_{A_\ell} \choose k_\ell} \bigg|  \Xi_A=N_1, \dots, \Xi_A=N_\ell \right]
= p^n  {N_1 \choose k_1} \cdots   {N_\ell \choose k_\ell} 
\end{align*}
 for any composition $\k \vdash n$  and for all disjoint Borel sets $A_1,\dots, A_\ell \subseteq \X$.
Hence, we obtain
\begin{align*}
\E\left[  {\widehat\Xi_{A_1} \choose k_1} \cdots   {\widehat\Xi_{A_\ell} \choose k_\ell} \right]
= p^n \E\left[  {\Xi_{A_1} \choose k_1} \cdots   {\Xi_{A_\ell} \choose k_\ell} \right] 
\end{align*}
so that,  by formula \eqref{correlation}, the correlation functions  of {\it the incomplete process} $\widehat{\Xi}$ are given by $p^n \rho_n(x_1,\dots , x_n)$ for all $n \ge 1$. 
In particular, we deduce from formula \eqref{det} that, if  $\Xi$ is a determinantal process with a correlation kernel $K$,  then the point process $\widehat{\Xi}$  is also determinantal with kernel $pK$.  

%  In particular, the  space  $\mathscr{Q}(\X)$ is equipped with its Borel sigma-algebra a natural notion of weak convergence. 






\section{Off-diagonal decay of the correlation kernel $K^N_V$ in dimension 2} \label{A:damping}

In this section, we review some classical estimates for the correlation kernel~\eqref{kernel} which  have been used in~\cite{AHM11} to prove the CLT \eqref{CLT_0}.  
Then, we prove lemma~\ref{thm:localization} and an analogous result for the cumulants of the \Ginibre process. For completeness, we also give the proof of lemma~\ref{thm:Ginibre_approximation}.  
We will use the formulation of section~5  in \cite{AHM11} but the estimates \eqref{exp_decay} and \eqref{tail_bound} go back to the papers~\cite{Berman09} and~\cite{AHM10}.  Suppose that the potential $V:\C \to\R$ is real-analytic and satisfies the condition \eqref{potential}. 
Then, there a function $\phi_V : \C \to \R^+$ such that $\phi_V(z) \ge \nu \log |z|^2$ as $|z| \to\infty$ and some constants $C, c, \delta>0$ such that
\begin{equation}\label{exp_decay}
\big| K_V^N(z,w) \big| \le C N e^{- c\sqrt{N} (|z-w|\wedge \delta )}  ,
\end{equation}
and 
\begin{equation}\label{tail_bound}
K_V^N(z,z) \le C N e^{-N  \phi_V(z)} , 
\end{equation}
for all $w \in \S$ and $z\in \C$. \\ % It was observed in~\cite{AHM11} that these estimates implies lemma~\ref{thm:localization}. 

{\it Proof of lemma~\ref{thm:localization}.} We use  the convention  $z_0=z_{n+1}$.  Since the kernel $K_V^N$ is reproducing, by the Cauchy-Schwartz  inequality, 
$$ \big| K_V^N (z,w) \big| \le \sqrt{ K_V^N (z,z) K_V^N (w,w) } , $$
for all $z, w\in\C$, so that
$$
\bigg| \prod_{j=0}^{n}K_V^N(z_j,z_{j+1}) \bigg|  \le \big| K_V^N(z_0,z_{1})  \big|\sqrt{K_V^N(z_0,z_{0})K_V^N(z_1,z_{1})} \prod_{j=2}^{n}K_V^N(z_j,z_j) .
$$

Since $\displaystyle \int_\C K_V^N(z,z) d\A(z) = N$ and $K_V^N(z,z)  \le C N $ (see the estimate \eqref{tail_bound}), we obtain 
$$
 \underset{|z_1-z_{n+1}|> \epsilon_N}{\int_{\C^n\times \S}}  \bigg|\prod_{j=0}^n K^N_V(z_{j}, z_{j+1})\bigg|\   d\A^{n+1}(\z)  \le  C N^n  \underset{|z_1-z_0|> \epsilon_N}{\int_{\S\times \C}} \hspace{-.2cm}  \big| K_V^N(z_0 ,z_1) \big| d\A(z_0)   d\A(z_1)  . 
$$
Then, it easy to check that the estimates  \eqref{exp_decay} and \eqref{tail_bound} imply that 
$$
 \underset{|z_1-z_0|> \epsilon_N}{\int_{\S \times \C}}   \big| K_V^N(z_0 ,z_1) \big| d\A(z_0)   d\A(z_1)  
\le C N  e^{-c \sqrt{N} \epsilon_N }  . 
$$
Hence, if $\epsilon_N = \kappa N^{-1/2} \log N $ and $\kappa \ge (n+1)/ c$, we obtain 
\begin{equation}\label{estimate_6}
 \underset{|z_1-z_{n+1}|> \epsilon_N}{\int_{\C^n \times \S}} \bigg| \prod_{j=0}^n K^N_V(z_{j}, z_{j+1})\bigg|\   d\A^{n+1}(\z)  = \O(N^{-1}) .
\end{equation}

Moreover, since  $\sup\big\{ |F_N(z_0, \z)| : \z \in \C^{n} , N  \ge N_0 \big\}  \le C \1_{z_0 \in\S}$ by  \eqref{estimate_0}, the estimate \eqref{estimate_6} implies that 
\begin{align} \label{estimate_4}
&\underset{z_0 = z_{n+1}}{\int_{\C^{n+1}}}\hspace{-.2cm} F_N(\z) \prod_{j=0}^n K^N_V(z_{j}, z_{j+1}) d\A^{n+1}(\z)  \\
&\notag\hspace{1cm}
 =  \underset{|z_1-z_{n+1}| \le \epsilon_N}{\int_{\C^n \times \S}} \hspace{-.2cm}  F_N(\z)   \prod_{j=0}^n K^N_V(z_{j}, z_{j+1}) \   d\A^{n+1}(\z)  + \O(N^{-1}) .
\end{align}

Now, we can proceed by induction to get formula \eqref{localization}. If
 $\mathscr{C}_N = \{\z \in \C^{n+1} :  z_{n+1} \in \S, |z_1-z_{n+1}| \le \epsilon_N \} $, the next step is to show that 
\begin{equation} \label{estimate_2}
 \underset{|z_2-z_1|> \epsilon_N}{\int_{\mathscr{C}_N}} \bigg| \prod_{j=0}^n K^N_V(z_{j}, z_{j+1})\bigg|\   d\A^{n+1}(\z)  = \underset{N\to\infty}{O}(N^{-1}) .
\end{equation}

Since the set $\S_V$ is open, there exists a compact set $\S' \subset \S_V$ such that $\S \subset \S'$ and  $\mathscr{C}_N \subset  \{\z \in \C^{n+1} :  z_{n+1} , z_1 \in \S' \}$  when the parameter $N$ is sufficiently large. Then, as before, we obtain
$$
 \underset{|z_2-z_{1}|> \epsilon_N}{\int_{\mathscr{C}_N}}  \bigg|\prod_{j=0}^n K^N_V(z_{j}, z_{j+1})\bigg|\   d\A^{n+1}(\z)  \le  C N^n  \underset{|z_2-z_1|> \epsilon_N}{\int_{\S'\times \C}} \hspace{-.2cm}  \big| K_V^N(z_1 ,z_2) \big| d\A(z_0)   d\A(z_1)  , 
$$
and  formula \eqref{estimate_2} also follows directly from the estimate \eqref{estimate_6}.
Hence, by formula \eqref{estimate_4}, this implies that
\begin{align*}
&\underset{z_0 = z_{n+1}}{\int_{\C^{n+1}}}\hspace{-.2cm} F_N(\z) \prod_{j=0}^n K^N_V(z_{j}, z_{j+1}) d\A^{n+1}(\z)  \\
&\notag\hspace{1cm}
 =  \underset{\begin{subarray}{c} |z_1-z_{n+1}| \le \epsilon_N  \\  |z_2-z_{1}| \le \epsilon_N  \end{subarray}} {\int_{\C^n \times \S}} \hspace{-.2cm}  F_N(\z)   \prod_{j=0}^n K^N_V(z_{j}, z_{j+1}) \   d\A^{n+1}(\z)  + \O(N^{-1}) .
\end{align*}
If we repeat this argument, we obtain \eqref{localization}. \qed\\


  
 \begin{lemma} \label{thm:delocalization}
Let $n \in\N$, $w_0= w_{n+1} =0$, and let $H(\w)$ be a polynomial of degree at most 2 in the variables $w_1,\dots, w_n, \bar{w_1},\dots, \bar{w_n}$.
 For any  sequence $\delta_N \ge k  \rho_N^{-1/2} \sqrt{\log\rho_N}$ with $k>0$, we have
\begin{align*}
& \int_{\C^n} H(\w) \prod_{j=0}^nK^\infty_{\rho_N}(w_{j}, w_{j+1})\  d\A^n(\w)  \\
  &\hspace{1cm}=  \int_{\mathscr{A}(0; \delta_N)} \hspace{-.2cm} H(\w) \prod_{j=0}^nK^\infty_{\rho_N}(w_{j}, w_{j+1})\   d\A^n(\w)  
+ \O(\rho_N^{1-k^2/2})
\end{align*}
where the set $\mathscr{A}(0; \delta_N)$ is given by formula \eqref{A}. 
 \end{lemma}
 
 \proof We will first  show that
\begin{equation} \label{estimate_1}
  \underset{|w_1| > \delta_N}{\int_{\C^n}}   H(\w) \prod_{j=0}^nK^\infty_{\rho_N}(w_{j}, w_{j+1}) \  d\A^n(\w)
   = \underset{N\to\infty}{O}\big( \rho_N e^{- \rho_N \delta_N^2/2}    \big) .
\end{equation}
 First, notice that if $H =1$, since $w_0=0$, by formula~\eqref{estimate_5}, we have 
  $$ 
  \underset{|w_1| > \delta_N}{\int_{\C^n}} \bigg| \prod_{j=0}^nK^\infty_{\rho_N}(w_{j}, w_{j+1})\bigg| \  d\A^n(\w) 
   \le  e^{- \rho_N \delta_N^2/2}    \rho_N^{n+1}
   \underbrace{\int_{\C^n} \prod_{j=1}^n e^{-\rho_N |v_j|^2/2}  d\A^n(\mathrm{v})}_{\displaystyle = \rho_N^{-n}} .    $$
 
In general, there exists a constant $C>0$ which only depends  on the polynomial $H$ so that 
$$
\big| H(\w) \big|  \le  C \big\{ 1+ |w_1|^2 +\cdots +|w_n|^2 \big\} 
$$
or 
  \begin{equation} \label{estimate_3}
\big| H(\w) \big|  \le  C \big\{ 1+ |w_2-w_1|^2 +\cdots +|w_{n-1} -w_n|^2 + |w_n|^2 \big\} . 
\end{equation}
Since, for any $k=1,\dots, n$,
\begin{align*}
&    \underset{|w_1| > \delta_N}{\int_{\C^n}}|w_k-w_{k+1}|^2 \bigg|  \prod_{j=0}^nK^\infty_{\rho_N}(w_{j}, w_{j+1})  \bigg| d\A^n(\w) \\
& \hspace{2cm}   \le  e^{- \rho_N \delta_N^2/2} 
   \underbrace{    \rho_N^{n+1} \int_{\C^n}  |v_k|^2 \prod_{j=1}^n e^{-\rho_N | v_{j} |^2/2}  d\A^n(\mathrm{v})}_{\displaystyle = 1} ,
   \end{align*}
the estimate \eqref{estimate_1} follows  directly from  \eqref{estimate_3} and the leading contribution comes from the constant term. If we use the estimate 
\begin{equation*} \big| H(\w) \big|  
\le  C \bigg\{ 1+ |w_1|^2 + \sum_{\begin{subarray}{c} j=1 \\ j \neq k \end{subarray}}^n |w_{j+1} -w_j|^2 +|w_n|^2 \bigg\} 
\end{equation*}
instead, the same argument shows that for any $k= 1,\dots n$,   
  \begin{equation*}
  \int\limits_{|w_k -w_{k+1}| > \delta_N } \hspace{-.3cm} H(\w) \prod_{j=0}^nK^\infty_{\rho_N}(w_{j}, w_{j+1}) \  d\A^n(\w)
   = \underset{N\to\infty}{O}\big( \rho_N e^{- \rho_N \delta_N^2/2}    \big) .
\end{equation*}
Hence, the lemma follows from applying a union bound and from the choice of the sequence $\delta_N$.  \qed\\
 
 





{\it Proof of lemma~\ref{thm:Ginibre_approximation}.}
The map $(z, w) \mapsto \Psi(z,w)$ is bi-holomorphic in a neighborhood of $(x_0, \bar{x_0})$, so there exists $0<\epsilon<1$ such that for all $|u|, |v| \le \epsilon$,
$$
\Psi(x_0 + u, \bar{x_0} + v) =
\sum_{k, j \ge 0} a_{kj} u^k v^j .
$$

By definition, 
$\overline{\Psi(z,w)} = \Psi(\overline{w},\overline{z})$, so that the coefficients of the previous power series are Hermitian-symmetric: $a_{kj} = \bar{a_{jk}}$ for all $k, j \ge 0$. Moreover, by definition,  we have
\begin{equation}\label{density}
a_{11} = \p_z \pb_z V |_{z=x_0} = \Delta V (x_0) = \frac{b_0(x_0, \bar{x_0})}{2} . 
\end{equation}

Let 
$$\h(u) = i \sum_{k>0} \big\{  a_{k0} u^k - a_{0k} \bar{u}^k \big\}   = -2  \Im\bigg\{  \sum_{k>0} a_{k0} u^k   \bigg\} . $$
 Since $V(x_0 + u) = \Psi(x_0 + u , \bar{x_0} + \bar{u} )$, we see that for any $|u|, |v| \le \epsilon$, 
\begin{align*}
& 2\Psi(x_0 + u, \bar{x_0} +\bar{v}) -V(x_0+u)- V(x_0+v)   \\
&\hspace{2cm}= - i\{ \h(u) - \h(v)\} +  a_{11} \big( 2 u \bar{v} - |\bar{u}|^2 -|\bar{v}|^2   \big)    
 + O( \epsilon^3) . 
\end{align*}
By formula \eqref{B_kernel}, this implies that  for any $|u|, |v| \le \epsilon_N = \log (N^\kappa) N^{-1/2}$, 
\begin{equation*}
\frac{B^N(x_0 + u , x_0 +v)e^{i N \h(u)}}{e^{i N \h(v)}}=  N b_0(x_0, \bar{x_0}) e^{N a_{11}  ( 2 u \bar{v} - |\bar{u}|^2 -|\bar{v}|^2)   } \left\{ 1 + \underset{N\to\infty}{O}\big(  (\log N)^{2}\epsilon_N   \big)  \right\} .
\end{equation*}
By formula \eqref{density} and the definition of the  \Ginibre kernel, it completes the proof. \qed\\

\end{appendices}


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{Lambert_a {\sc G.~Lambert, {\em Mesoscopic fluctuations for unitary invariant ensembles. \newblock arXiv:1510.03641.
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{Pastur06 {\sc L.A.~Pastur, {\em Limiting laws of linear eigenvalue statistics for {Hermitian matrix models, J. Math. Phys. 47, (2006).
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{PS11 {\sc L.A. Pastur and M.~Shcherbina, {\em Eigenvalue {Distribution of {Large {Random {Matrices, Mathematical Surveys and Monographs 171, Amer. Math. Soc., Providence, RI, 2011.
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{RV07a {\sc B.~Rider and B.~Vir{{\'ag, {\em The noise in the circular law and the {Gaussian free field, Int. Math. Res. Not. IMRN (2007).
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{Shcherbina13 {\sc M.~Shcherbina, {\em Fluctuations of linear eigenvalue statistics of $\beta$-matrix models in the multi-cut regime, J. Stat. Phys. 151, (2013), 1004--1034.
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{Soshnikov01 {\sc A.~Soshnikov, {\em Gaussian limit for determinantal random point fields, Ann. Probab. 30, (2001), 1--17.
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{Spencer11 {\sc T.~Spencer, {\em Random banded and sparse matrices in {\em The Oxford handbook of random matrix theory, Oxford Univ. Press, Oxford, 2011, 471--488.
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