Comportement asymptotique de la d\'ecomposition spectrale du noyau de transition d'une marche al\'eatoire sur $\Sy_n$
Antoine Dahlqvist
Antoine
We show that the Laplace transforms of traces of words in independant unitary Brownian motions converge towards an analytic function on a non trivial disc. This results allow to study asymptotics of Wilson loops under the unitary Yang-Mills measure on the plane. The limiting objects obtained are shown to be characterized by equations analog to Schwinger-Dyson's ones, named here after Makeenko and Migdal.
BibFY.tex
\def\cprime{$'$}
\begin{thebibliography}{10}

\bibitem{ASMF}
Michael Anshelevich and Ambar~N. Sengupta.
\newblock Quantum free yang-mills on the plane.
\newblock {\em Journal of Geometry and Physics}, 62(2):330 -- 343, 2012.

\bibitem{BP}
Thomas~F. Banchoff and William~F. Pohl.
\newblock A generalization of the isoperimetric inequality.
\newblock {\em J. Differential Geometry}, 6:175--192, 1971/72.

\bibitem{TCLF}
Florent Benaych-Georges.
\newblock Central limit theorems for the {B}rownian motion on large unitary
  groups.
\newblock {\em Bull. Soc. Math. France}, 139(4):593--610, 2011.

\bibitem{BM}
Olivier {Bernadi} and Alejandro~H. {Morales}.
\newblock {Counting trees using symmetries}.
\newblock {\em Journal of Combinatorial Theory Serie A}, 123(1):104--122, 2014.

\bibitem{BianeMBL}
Philippe Biane.
\newblock Free {B}rownian motion, free stochastic calculus and random matrices.
\newblock In {\em Free probability theory ({W}aterloo, {ON}, 1995)}, volume~12
  of {\em Fields Inst. Commun.}, pages 1--19. Amer. Math. Soc., Providence, RI,
  1997.

\bibitem{BianeSegalBarg}
Philippe Biane.
\newblock Segal-{B}argmann transform, functional calculus on matrix spaces and
  the theory of semi-circular and circular systems.
\newblock {\em J. Funct. Anal.}, 144(1):232--286, 1997.

\bibitem{BorotGuionnet}
G.~Borot and A.~Guionnet.
\newblock Asymptotic expansion of {$\beta$} matrix models in the one-cut
  regime.
\newblock {\em Comm. Math. Phys.}, 317(2):447--483, 2013.

\bibitem{BIPZ}
E.~Br{\'e}zin, C.~Itzykson, G.~Parisi, and J.~B. Zuber.
\newblock Planar diagrams.
\newblock {\em Comm. Math. Phys.}, 59(1):35--51, 1978.

\bibitem{CK}
G.~{C{\'e}bron} and T.~{Kemp}.
\newblock {Fluctuations of Brownian Motions on GL\_N}.
\newblock {\em ArXiv e-prints}, September 2014.

\bibitem{Collins}
Beno{\^{\i}}t Collins.
\newblock Moments and cumulants of polynomial random variables on unitary
  groups, the {I}tzykson-{Z}uber integral, and free probability.
\newblock {\em Int. Math. Res. Not.}, (17):953--982, 2003.

\bibitem{CollinsGuionnetSegala}
Beno{\^{\i}}t Collins, Alice Guionnet, and Edouard Maurel-Segala.
\newblock Asymptotics of unitary and orthogonal matrix integrals.
\newblock {\em Adv. Math.}, 222(1):172--215, 2009.

\bibitem{HOPS}
Beno{\^{\i}}t Collins, James~A. Mingo, Piotr {\'S}niady, and Roland Speicher.
\newblock Second order freeness and fluctuations of random matrices. {III}.
  {H}igher order freeness and free cumulants.
\newblock {\em Doc. Math.}, 12:1--70 (electronic), 2007.

\bibitem{IntBM}
A.~{Dahlqvist}.
\newblock {Integration formula for Brownian motion on classical compact Lie
  groups}.
\newblock {\em ArXiv e-prints}, December 2012.

\bibitem{DiaconisEvans}
Persi Diaconis and Steven~N. Evans.
\newblock Linear functionals of eigenvalues of random matrices.
\newblock {\em Trans. Amer. Math. Soc.}, 353(7):2615--2633, 2001.

\bibitem{Diashah}
Persi Diaconis and Mehrdad Shahshahani.
\newblock On the eigenvalues of random matrices.
\newblock {\em Jour. Appl. Prob.}, 31(3):49--62, 1994.

\bibitem{Driver}
Bruce~K. {Driver}.
\newblock {YM$\sb 2$: Continuum expectations, lattice convergence, and lassos.}
\newblock {\em {Commun. Math. Phys.}}, 123(4):575--616, 1989.

\bibitem{AFM}
V.~{F{\'e}ray}, P.-L. {M{\'e}liot}, and A.~{Nikeghbali}.
\newblock {Mod-phi convergence and precise deviations}.
\newblock {\em ArXiv e-prints}, April 2013.

\bibitem{Franck}
Franck Gabriel.
\newblock Planar markovian holonomy fields : construction, characterization and
  application.
\newblock {\em Paper in preparation}.

\bibitem{GrossMat}
David~J. Gross and Andrei Matytsin.
\newblock Some properties of large-{$N$} two-dimensional {Y}ang-{M}ills theory.
\newblock {\em Nuclear Phys. B}, 437(3):541--584, 1995.

\bibitem{NovakGuionnet}
A.~{Guionnet} and J.~{Novak}.
\newblock {Asymptotics of unitary multimatrix models: The Schwinger-Dyson
  lattice and topological recursion}.
\newblock {\em ArXiv e-prints}, January 2014.

\bibitem{HC}
Harish-Chandra.
\newblock Differential operators on a semisimple {L}ie algebra.
\newblock {\em Amer. J. Math.}, 79:87--120, 1957.

\bibitem{HUMPHRIES}
Stephen~P. Humphries.
\newblock On weakly distinguished bases and free generating sets of free
  groups.
\newblock {\em The Quarterly Journal of Mathematics}, 36(2):215--219, 1985.

\bibitem{IkedaWatanabe}
Nobuyuki Ikeda and Shinzo Watanabe.
\newblock {\em Stochastic differential equations and diffusion processes},
  volume~24 of {\em North-Holland Mathematical Library}.
\newblock North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd.,
  Tokyo, 1981.

\bibitem{KazakovMF}
V.A. Kazakov.
\newblock Wilson loop average for an arbitrary contour in two-dimensional u(n)
  gauge theory.
\newblock {\em Nuclear Physics B}, 179(2):283 -- 292, 1981.

\bibitem{KazakovKostov}
V.A. Kazakov and I.K. Kostov.
\newblock Computation of the wilson loop functional in two-dimensional
  u$(\infty)$ lattice gauge theory.
\newblock {\em Physics Letters B}, 105(6):453 -- 456, 1981.

\bibitem{ThierrySW}
Thierry L{\'e}vy.
\newblock Schur-{W}eyl duality and the heat kernel measure on the unitary
  group.
\newblock {\em Adv. Math.}, 218(2):537--575, 2008.

\bibitem{MF}
Thierry {L{\'e}vy}.
\newblock {The master field on the plane}.
\newblock {\em ArXiv e-prints}, December 2011.

\bibitem{ThierryMylene}
Thierry L{\'e}vy and Myl{\`e}ne Ma{\"{\i}}da.
\newblock Central limit theorem for the heat kernel measure on the unitary
  group.
\newblock {\em J. Funct. Anal.}, 259(12):3163--3204, 2010.

\bibitem{Champsmarkoholo}
Thierry L�vy.
\newblock {\em Two-dimensional Markovian holonomy fields.}
\newblock Ast�risque 329, 2010.

\bibitem{MM}
Yu.M. Makeenko and A.A. Migdal.
\newblock Exact equation for the loop average in multicolor qcd.
\newblock {\em Physics Letters B}, 88:135 -- 137, 1979.

\bibitem{PLM}
P.-L. {M{\'e}liot}.
\newblock {The cut-off phenomenon for Brownian motions on symmetric spaces of
  compact type}.
\newblock {\em ArXiv e-prints}, October 2012.

\bibitem{SenguptaYMCompact}
Ambar Sengupta.
\newblock Gauge theory on compact surfaces.
\newblock {\em Mem. Amer. Math. Soc.}, 126(600):viii+85, 1997.

\bibitem{IS}
Isadore~Manuel Singer.
\newblock In Simon Gindikin, James Lepowsky, and Robert~L. Wilson, editors,
  {\em Functional analysis on the eve of the 21st century. {V}ol. {I}}, volume
  131 of {\em Progress in Mathematics}, page 263. Birkh\"auser Boston Inc.,
  Boston, MA, 1995.
\newblock In honor of the eightieth birthday of I. M. Gel{\cprime}fand, Papers
  from the conference held at Rutgers University, New Brunswick, New Jersey,
  October 24--27, 1993.

\bibitem{Hooft}
G.~{'t Hooft}.
\newblock {A planar diagram theory for strong interactions}.
\newblock {\em Nuclear Physics B}, 72:461--473, April 1974.

\bibitem{Xu}
Feng Xu.
\newblock A random matrix model from two-dimensional {Y}ang-{M}ills theory.
\newblock {\em Comm. Math. Phys.}, 190(2):287--307, 1997.

\bibitem{ZZ}
P.~Zinn-Justin and J.-B. Zuber.
\newblock On some integrals over the {${\rm U}(N)$} unitary group and their
  large {$N$} limit.
\newblock {\em J. Phys. A}, 36(12):3173--3193, 2003.
\newblock Random matrix theory.

\end{thebibliography}

FY.tex

\documentclass[12pt]{amsart}
\usepackage{geometry} % see geometry.pdf on how to lay out the page. There's lots.
\geometry{a4paper} % or letter or a5paper or ... etc
% \geometry{landscape} % rotated page geometry
\pdfoutput=1 






% See the ``Article customise'' template for come common customisations
\usepackage[applemac]{inputenc}
\input{MacArxiv}

\usepackage{color}
\usepackage{amsfonts}
\usepackage{amsmath,amssymb,amsthm}% Formules math�matiques

\usepackage{enumerate}
%\usepackage{pdftricks}
\usepackage{graphicx}
%\definecolor{}{cmyk}{1,0,0,0}
\usepackage[citecolor=cyan,colorlinks=true,linkcolor=blue]{hyperref}


\title[Free energies and fluctuations for  the unitary Brownian motion]
{Free energies and fluctuations for  the unitary Brownian motion}

\address{Technische Universit�t Berlin, Institut f�r Mathematik, RTG 1845, D-10623 Berlin, Germany}
\author{Antoine Dahlqvist}
\thanks{Department of Mathematics, TU Berlin, \emph{E-mail address:} dahlqvis@math.tu-berlin.de }




%\bibliographystyle{alpha}
%\bibliography{unitary-final}

\begin{document}



\begin{abstract}We show that the Laplace transforms of  traces of words in independant unitary Brownian motions converge towards an analytic function on a non trivial disc.  This results allow to study  asymptotics of Wilson loops under the unitary Yang-Mills measure on the plane. The limiting objects obtained are shown to be characterized  by equations analog to Schwinger-Dyson's ones, named here after Makeenko and Migdal. 
\end{abstract}




\maketitle



\section{Introduction}

The following paper aims at studying traces of non-commutative  polynomials in independant Brownian motions on the  group of unitary matrices $\Un(N),$ as the  size $N$ goes to infinity.   In \cite{BianeMBL, Xu, ThierrySW, MF}, it has been shown that for Brownian motions invariant by conjugation, with a proper time-scale, these  traces, properly normalized, converge towards a deterministic limit given by the evaluation of the free Brownian motion.  We want here to study the Laplace transform of these random variables with normalization analog to one of the mod-$\phi$ convergence (\cite{AFM}). As a corollary, we obtain the fluctuations around their  limit.  In \cite{ThierryMylene}, the fluctuation of traces in polynomials of one marginal were given, the latter point of the present work gives an extension of their result. Simultaneously to the writing of the present paper, G. C�bron and T. Kemp obtained in \cite{CK} a similar result of gaussian fluctuation for diffusions on $\GL_N(\C)$, among which the $\Un(N)$ Brownian motion  is a special case. A second motivation of our paper is to study the planar Yang-Mills measure for large unitary groups. We are able here to show the convergence to all orders of the Wilson loops and prove that the limiting objects are caracterized by analog of Schwinger-Dyson equations. In particular, we prove  the existence of a gaussian field indexed by rectifiable loops describing the fluctuation of the convergence towards the master field proved in \cite{MF}.  


\vspace{0,5 cm}
\textbf{Free energies of matricial models:} In many random matricial models,  asymptotics of $\esp[e^{N\Tr(V)}],$ where $V$ is a fixed non-commutative polynomial in a sequence of random matrices of size $N$,  have  been extensively studied and have several applications ranging from theorical physics, through enumerative combinatorics, free probability and representation theory. A case of study is the Harisch-Chandra-Itzykon-Zuber integral (\cite{HC, ZZ}) 
$$H(A,B)=\esp[e^{N\Tr(AUBU^*)}],$$
where $A$ and $B$ are two deterministic hermitian matrices and $U$ is a random unitary matrix, distributed according to the Haar measure.
 When the non-commutative polynomial plays the role of the potential of  a Gibbs measure, the  normalized logarithm of Laplace tranforms is called the free energy and have been  studied in several places, for example in  \cite{NovakGuionnet, CollinsGuionnetSegala, BorotGuionnet}.  In the pioneering work  \cite{BIPZ},  formal  expansions have been proposed for several physical models. In \cite{Collins, HOPS}, technics have been developed to study  formal expansions for model of random matrices with properties of invariance by conjugation.  We shall give here a converging expansion for the following model.  


Let $(U_{1,t_1},\ldots,U_{q,t_q})_{t\in \R_+^q}$ be  $q$ independant Brownian motions invariant by adjunction in $\Un(N)$ (see section 
\ref{section def} for a definition) and denotes by $\Tr$ the usual non-normalized trace of matrices.
\begin{theom} \label{Theom Energy Words}For $t\in\R_+^q $  and any non-commutative polynomial $V$ in $2q$ variables, there exists 
$r_V>0$ and analytic functions $\vp_{t,V}, (\psi_{t,V,N})_{N\ge 1}$ and $\psi_{t,V}$ on $D_{r_V}=\{z\in\C: |z|<r_V\},$ such that 
$$ e^{\psi_{t,V,N}(z)}=\esp[e^{zN\Tr(V(U_{i,t_i},U_{i,t_i}^{*}, i=1..q))- N^2\vp_{t,V}(z)}]\longrightarrow e^{\psi_{t,V}(z)},$$
as $N\to\infty,$ where the convergence is uniform on compact subset of $D_{r_V}$.\end{theom}
For any non-commutative polynomial $V$ in $2q$ variables, whose evaluation is hermitian one unitary matrices, we shall define for any integer $N\ge 1,$ a probability measure $\mu_{N,V}$ on $\Un(N)^{q}$ absolutly continuous with respect to the law of $(U_{i,t_i})_{1\le i \le q}$, with density proportional to  $e^{zN\Tr(V(U_{i,t_i},U_{i,t_i}^{*}, i=1..q))}.$  Then, for any $N\ge 1,$  $(U^V_{N,1}, \ldots,U_{N,q}^V)  $ denotes a random variable with law $\mu_{N,V}.$
 \begin{theom}\label{MF Potential}  If $V,W\in \C\la X_i,Y_i\ra_{i=1..q}$ are  non-commutative polynomials  with  small enough coefficients and $V^*=V$, then,  under the probability measure $\mu_{N,V}$, the random variable $\frac{1}{N}\Tr(W(U^V_i,{U^V_i}^*, i=1..q))$ converges in probability towards a constant  $\Phi_{t,V}(W).$ 
\end{theom}

%if  the non-commutative polynomial satisfies $V(X_1,\ldots,X_q,Y_1,\ldots,Y_q )= \overline{V
 %(Y_1,\ldots,Y_q ,X_1,\ldots,X_q)}$, then the function $I_{t,V,N}(x)=N^{-2}\log \esp [e^{xN \Tr(V(U_{i,t_i},U_{i,t_i}^{*}, i=1..n))}] $ converges uniformly  towards $\vp_{t,V}(x),$ on compact intervals of  $]-r_V,r_V[$.




\textbf{Yang-Mills measure  on the plane:} We shall see that this result can be partly extended to the framework of Yang-Mills measure that has been developed in \cite{Driver, SenguptaYMCompact, ASMF, Champsmarkoholo, MF}. Therein, we give a recursive way to compute coefficients of $\vp_{t,V}(z)$,  proving analogs of Schwinger-Dyson equations, called here Makeenko-Migdal equations. The latter equations for the first coefficient in $z$ appeared in \cite{MM}  and were first  proved rigorously in \cite{MF}.  The Yang-Mills measure  encompasses the different models for  all $q\in\N^*$  and $ t\in \R_+^q,$ into one random object, for which the recursive equations has  a  simple  interpretation.   We shall use the approach of \cite{Champsmarkoholo,MF, Franck} by considering for any $N\ge 1,$ a process $(H_l)_l$ indexed by the set $\Ld(\R^2)$ of rectifiable loops in the plane, valued in $\Un(N)$, whose law will be denoted by  $YM_N$.

\vspace{0,5 cm}

\textbf{Planar master field:} The works  \cite{ASMF, MF}  proved that under $YM_N$, the random field $(\frac{1}{N}\Tr(H_l))_{l\in\Ld(\R^2)}$ converges in probability towards a deterministic field $(\Phi(l))_{l\in \Ld(\R^2)}$. The statement of this result first appeared in the physics literature, in the study of QCD,   with the  works \cite{KazakovMF, KazakovKostov, MM}, and in the mathematical paper \cite{IS}, as a conjecture. The limiting field was named therein  \emph{master field}, following the terminology of  \cite{Hooft}.  This object is  the first coefficient of the extension of the functions $\vp_{t,0}$ in Theorem \ref{Theom Energy Words}.  The asymptotic of $2D$-Yang-Mills   measure on other compact surfaces has also been investigated in the physics literature  \cite{GrossMat}.  It  won't be discussed in this text but could lead to future works.


\vspace{0,5 cm}

\textbf{Fluctuations:} The study of fluctuation of traces of random elements of a compact group of large dimension started with \cite{Diashah}, where is was investigated, thanks to representation theory tools, for the Haar measure on the classical compact Lie groups. The Theorem \ref{Theom Energy Words} allows us in particular to characterize the fluctuations in the convergence of the non-commutive distribution of a $\Un(N)$-Brownian motion towards the free unitary Brownian motion distribution.  We further prove that  under $YM_N$ the random field $\left(\Tr(H_l)- \esp[H_l]\right)_{l\in \Ld(\R^2)}$ converges in law towards a  gaussian field $(\phi_l)_{l\in \Ld(\R^2)}$, characterized by the Makeenko-Migdal equations.  Besides, we observe that when the loops are dilated by a factor $\lambda$, the above  fields have the same gaussian behavior as $\lambda\to 0$. Our result  extends the work of \cite{ThierryMylene}, which study the gaussian fluctuations in  the convergence  of the empirical measure of a $\Un(N)$-Brownian motion marginal. The gaussian process obtained therein can be shown to be a deformation the one obtained in \cite{Diashah}.   The fluctuation results presented in this text are extracted from the PhD thesis of the author, where the case of the orthogonal and  symplectic groups have also been addressed.  Note also that in \cite{DiaconisEvans,TCLF},  fluctuations with another scaling are studied:  the one of finite block of a random matrix.    For simplicity, we shall restrict here to  the study of traces of words in the unitary case.  \vspace{0,5 cm}

\textbf{Organisation of the paper:} The next section is devoted to the description of the convention we use  for the standard Brownian motion on  $\Un(N)$ and  the choice of scaling we made.   In  sections 3 and 4 ,  are obtained the main expressions and estimates needed to get our result. In sections 5 and 6, we give their applications to study respectively the unitary Brownian motion and the Yang-Mills measure. In the last section, we show that the limited object obtained in the paper can be characterized by the recursive equations of Makeenko and Migdal. 

\tableofcontents

%The aim of this text is to study the  fluctuation result for the convergence of the unitary Yang-Mills measure towards the master field on the plane proved in \cite{MF}.  The limiting object is a Gaussian field indexed by rectifiable paths on the plane. During the  proof, we get  that the Yang-Mills measure converges to all orders in the sense of higher order free probability.  These limits define invariants for $k-$tuples of   loops of finite length that match at the first order the $\Un(1)$-Yang-Mills as the area goes to zero. We give  a combinatorial proof of traces fluctuations for the Brownian motion and    a simple expression for the generating function of their covariances. We give also a combinatorial proof of the Makeenko-Migdal equations for the unitary group.



\section{Unitary Brownian motion and its large \texorpdfstring{$N$}{Lg} limit \label{section def}}




%In this section, we shall lay the corner stone to prove our theorem. We will first study  traces of    powers of one single unitary Brownian motion.   We will present the results obtained in \cite{BianeMBL} and \cite{ThierryMylene} about convergence and fluctuations of these functions as $N\to\infty$. We give a different  proof of the fluctuations: we shall prove that the unitary Brownian motion converges to all orders which implies the Gaussian fluctuations of  traces of polynomials in the unitary Brownian motion.  Though, we do not prove the fluctuation for Lipschitz functions  proved in \cite{ThierryMylene} but focus on polynomials. What is more, we give a simple expression for the generating series of the covariances. One application of this formula is to give easily the behavior of covariances as $t\to \infty.$ 


\subsection{Definition and time scale of unitary Brownian motion}  For any integer $N$, we shall write $\Un(N)$ for  the group of unitary matrices of $M_N(\C)$ and $\uN$ for its Lie algebra, that is,  the set of skew-hermitian matrices. We define a scalar product   $\langle\cdot, \cdot\rangle$  on $\uN$  by setting for any $X,Y\in \uN,$
 $$\langle X,Y \rangle= - N \Tr(XY).$$
Let us write $(K_t)_{t\ge 0}$ the Brownian motion on the Euclidean space $(\uN, \langle \cdot,,\cdot  \rangle)$ and recall that it is a Gaussian process such that for any $X,Y\in \uN, t,s\ge0$, $$\esp[ \langle X, K_t \rangle  \langle Y,  K_s\rangle]= \langle X ,Y\rangle \min (t,s).$$ 
Let us define $(U_t)_{t\ge 0}$ as the $M_N(\C)$-valued  solution of the following stochastic differential equation: 
\begin{align*}\label{EDSU}dU_t &=  U_tdK_t -\frac{1}{2}U_t dt   \tag{*}   \\
U_0  &= \Id.    
\end{align*}
\begin{lemma}i) Almost surely, for all $t\ge 0,$ $U_t\in \Un(N)$.

ii)  For all $T\ge 0,$  $(U_T^*U_{T+t})_{t\ge 0}$ is independant of the sigma field $\sigma(U_s, s\le T)$ and has the same law as $(U_t)_{t\ge 0}$.

iii) For any $t\ge 0$ and every fixed $U\in \Un(N),$ $UU_tU^{-1}$ has the same law as $U_t.$
\end{lemma}
\begin{proof} Let us prove the first point, the two others are left to the Reader. The processes $(i \sqrt{N} (K_t)_{p,p})_{t\ge 0}$ for $1\le p\le N$ and $(\sqrt{N}(K_t)_{i,j})_{t\ge 0}$ for  $1\le i<j\le N$   are $N^2$ independant processes,  the $N$ first have the same law as a standard real Brownian motions, whereas the others are distributed as standard complex Brownian motions, so that $\esp[|(K_1)_{1,2}|^2]=1.$ Let us denote by  $\la\!\langle \cdot  \ra\!\rangle$  the symbol of quadratic variations, so that 
  $$\la\!\la dK_t. dK_t\ra\!\ra =\sum_{1\le i,p,j\le N}\la\!\la  d(K_t)_{i,p} d(K_t)_{p,j}\ra\!\ra E_{i,j}= - dt \Id. $$
It�'s formula then yields
 $$d\left(U_t U_t^*\right)=  U_t (dK_t+dK_t^*)U_t^* +U_t (\la\!\la dK_t. dK_t^*\ra\!\ra- dt \Id )U_t^*=0.  $$\end{proof}
\noindent We call  this process  the \emph{$\Un(N)$-Brownian motion}\footnote{It can be shown that it is a diffusion on the Riemannian manifold $\Un(N)$ endowed by the left-invariant metric associated to $\la\cdot,\cdot\ra$ and that its generator is the Laplace-Beltrami operator (see \cite{IkedaWatanabe} or \cite{ThierryMylene}, Proposition 2.1. for an elementary proof).} (see \cite{IntBM,MF} for a similar definition on other classical compact groups). For $N=1$, it has the same law as $(e^{iB_t})_{t\ge 0}$, where $(B_t)_{t\ge 0}$ is the standard real Brownian motion. Let us make  remarks on the scaling.  Let us recall that  the scalar product $\langle \cdot,\cdot \rangle$  on $\uN$ induces a Riemannian metric $d$ on $\Un(N)$. On the one hand,  this choice of metric  yields that the diameter of $\Un(N)$ is $d(\Id,-\Id)=\int_0^1 \|\dot{\gamma}_t\|dt,$ where $\gamma:t\in[0,1]\mapsto \exp(t i\pi \Id_N )$, that is, $ \| i\pi  \Id \| =  N\pi$. On the other hand,  the law of large numbers implies that $\dim( \uN )^{-1} \| K_t \|^2=N^{-2} \| K_t\|^2  $ converges, as $N\to \infty$, towards $ t$. Heuristically, we may infer that, as $N\to\infty$, for any $t>0$, $d(\Id, U_t)$ behaves like $\|K_t\|$  and   $\frac{d(U_t,  \Id)}{d(\Id,-\Id)}\to C_t\in (0,\infty)$. With this scaling, the Brownian motion �has the time to visit''\footnote{Note that  a good scaling to study the convergence of the distance in total variation $d_{TV}$ between  the law of Brownian motion and the Haar measure, is  faster than ours. Let $U$ be a Haar distributed random variable on $\Un(N)$. It has been  shown in \cite{PLM} that  the  function $t\mapsto d_{TV}(U_{t\log(N)},U)$  admits a cut-off  around the value $t=2$. } $\Un(N).$   Besides,   the stochastic differential equation (\ref{EDSU}) does not depend on $N$ and  such an equation makes sense in the context of free stochastic differential equations (see \cite{BianeMBL}).  Let us add a last comment on the time-scale.  With the above choice, the $\Un(1)$-Brownian motion appears with the same scaling in all $\Un(N)$-Brownian motions.
\begin{lemma} For any $N\in \N^*$, let $(U_{t,N})_{t\ge 0}$ be a $\Un(N)$-Brownian motion. Then,  the process $(\det(U_{t,N}))_{t\ge 0}$ has the same distribution as $(\Un_{t,1})_{t\ge 0}$.\label{det}
\end{lemma} 
\begin{dem} Observe that for any $N\in\N^*$, $(i\Tr(K_t))_{t\ge 0}$  has the same law as a standard Brownian motion. If $D_2(\det)_M:M_N(\C)^2\to\C$ denotes the second derivative of the determinant at a point $M\in M_N(\C)$, It�'s formula yields that $$d \left(\det(U_t)\right)= \det(U_t) d\Tr(K_t)-\frac{N}{2} \det(\Un_t)dt+\frac{\det(U_t)}{2}\la\!\la D_2(\det)_{\Id}(dK_t,dK_t)\ra\!\ra.$$
What is more,
\begin{align*}
\la\!\la D_2(\det)_{\Id}(dK_t,dK_t)\ra\!\ra &=\sum_{1\le i<j\le N}\left(\la\!\la (dK_t)_{i,i} ,(dK_t)_{j,j} \ra\!\ra-\la\!\la (dK_t)_{j,i} ,(dK_t)_{i,j} \ra\!\ra\right)\\
&=- \frac{N(N-1)}{2N}dt.
\end{align*}
Hence, $$d\left(\det(U_t)\right)=\det(U_t)d\left(\Tr(K_t)\right)-\frac{1}{2}\det(U_t)dt$$
and $\det(U_0)=1.$
\end{dem}
\subsection{Free unitary  Brownian motion\label{LimitdistUBM}}
Let us recall the first result obtained about the behavior of unitary Brownian motion in large dimension.  We shall denote by $(\mu^N_t)_{t\ge 0}$  the family of random measures given by the empirical measure of eigenvalues of $\Un_t$: if $\lambda_1,\ldots,\lambda_N \in \U$ are the eigenvalues of $\Un_t$, $\mu_t^N=\frac{1}{N}\left(\delta_{\lambda_1}+\cdots +\delta_{\lambda_N}\right)$.  For any polynomial function $P\in\C[X]$,  note that $\tr ( P(\Un_t))=\int_{\U} P(z)\mu_t(dz)$.  The following theorem has first been proved in \cite{BianeMBL} using harmonic analysis on the unitary group.
\begin{theom}[\cite{BianeMBL,Xu,ThierrySW}] \label{theoBiane}The sequence or random measures $(\mu_t^N)_{N\ge 0}$ converges   weakly in probability\footnote{\label{notetopo}We mean here that for any continuous function $f$, the sequence of random  variables $(\int f d\mu^N_t)_{N\ge 1}$ converges in probability to the constant $\int f d\mu_t$.},  towards a deterministic measure $\mu_t$ on $\U$, whose moments are given as follows: $$\mu_{t,n}= \int_{\U} z^n \mu_t(dz) = e^{-\frac{nt}{2}} \sum_{k=0}^{n-1} \frac{(-t)^k}{k!} n^{k-1}{n\choose k+1} .$$
\end{theom} 
Using the property of independance and stationarity of multiplicative increments a $\Un(N)$-Brownian motion, together with the invariance of its law by  adjunction and free probability arguments, the following Theorem was then deduced.
\begin{theom}[\cite{BianeMBL}] For any $t_1,t_2,\ldots,t_q\ge 0$ and $V$ any non-commutative polynomial of $2q$-variables, the random variables
$ \frac{1}{N}\Tr(V(U_1,U_1^*,\ldots,U_q,U_q^*)) $ converge in probability towards a constant.
\end{theom}
The limiting object is called the non-commutative distribution of the \emph{free unitary brownian motion} and can be characterized by  the family of measures $(\mu_t)_{t\ge 0}$ together with the asymptotic freeness of the increments. This last theorem was proved  in another way in \cite{Xu, ThierrySW,MF} showing directly  the convergence for any non-commutative polynomial.  Let us recall how the argument of \cite{ThierrySW, MF} goes to show \ref{theoBiane}.  Let us denote by $\Sy_n$ the group of permutations of $\{1,\ldots,n\}.$ For any permutation $\sigma\in \Sy_n$ composed of  $\# \sigma$ cycles, we define a function $f_\sigma$ on $\Un(N)$  by setting  for any $U\in\Un(N),$
$$ f_\sigma (U)= N^{-\# \sigma}\Tr \left(\sigma U^{\ts n} \right)$$
and a function on $\Sy_n$, by setting for any $t>0$,
$$\varphi_t^N(\sigma)=\esp[f_\sigma( U_{N,t})].$$ 
Then, the latter family of functions on the symmetric group is shown to satisfy the following  differential system (see \cite{ThierrySW} or Lemma \ref{EDPTens}).
\begin{lemma}[\cite{ThierrySW}] \label{lemEDPpol} For any permutation $\sigma\in\Sy_n$,  
\begin{align*}\frac{d}{dt} \varphi_t^N(\sigma) &= -\frac{n}{2} \varphi_t^N(\sigma) - \sum_{1\le i<j\le n}N^{\# \sigma (i  \hspace{0,1 cm} j)-\# \sigma-1} \varphi_t^N(\sigma (i \hspace{0,1cm} j)),\\
\varphi_0^N(\sigma)&=1.
\end{align*} 
\end{lemma} 

The unique solution of this  system of ordinary differential equations  is a  power series  in $\frac{1}{N}$ and converges, as  $N\to\infty$, to a function $\varphi_t.$ It can further be shown to satisfy for any  $\sigma \in \Sy_n$ with $a_k$ cycles of length $k$, 
$$\varphi_t(\sigma)= \prod_{k=1}^n \varphi_t( ( 1 \cdots k))^{a_k}.$$
Setting for all $t\ge 0, n\ge1$, $\mu_{t,n}=  \varphi_t(( 1 \cdots  n))$, the limit in $N$  of the former equations takes the following form: 
\begin{equation} \frac{d}{dt}\mu_{t,n}= -\frac{n}{2}\mu_{t,n} - \frac{n}{2}\sum_{k=1}^{n-1} \mu_{t,k}\mu_{t,n-k}, \label{cut}
\end{equation} 
with initial condition $\mu_{0,n}=1.$  This system of equations is then shown to have as unique solution given by the expression of Theorem \cite{BianeMBL}. It  follows that for $n\in\N, t\ge 0$, $$\esp[\int_\U \omega^n\mu_t^N(d\omega) ]\to \mu_{t,n}. $$ To conclude and obtain a convergence in probability, one ultimately needs to estimate the covariances of the complex variables $\left(\frac{1}{N}\Tr(U_t^n)\right)_{n\in \N, t\ge 0}$  with their complex conjugate.  This latter point together with the Lemma \ref{lemEDPpol} can be  proved using the following lemma, that allows to study  any polynomial in the entries and their conjugate of a unitary Brownian motion.

\vspace{0,5 cm}

 For any integer $n\in\N^*$, let us recall the left action of $\Sy_n$ on $\C^{\ts n},$ such that  for any permutation $\sigma\in \Sy_n$  and any elementary tensor  ${v_1}\ts v_2\ts \ldots \ts v_n\in (\C^N)^{\ts n},$
$$\sigma. {v_1}\ts v_2\ts \ldots \ts v_n= v_{\sigma^{-1}(1)}\ts \ldots v_{\sigma^{-1}(n)}. $$
The endomorphism of  $(\C^N)^{\ts n}$ associated to a permutation $\sigma$ will be abusively denoted below by the same symbol.
For any pair of distinct integers $i,j\in \{1,\ldots ,n\}$, we denote by $ \langle i\hspace{0,1cm} j\rangle$  the endomorphism of $(\C^N)^{\ts n}$  which acts like the  endomorphism $\sum_{1\le  r,s \le N}E_{r,s}\ts E_{r,s}$ on the $i$th  and $j$th  tensors  and trivially on the others.    

\begin{lemma}[\cite{IntBM, MF}] \label{EDPTens}Let $U_{t}$ be a  Brownian motion on $\Un(N)$. For any positive integers  $a,b$, which add up to $n$, the following differential equation holds: 

$$\frac{d}{dt}\esp[U_t^{\ts a}\ts  \overline{U}_t^{\ts b }]= -\esp[U_t^{\ts a}\ts \overline{U}_t^{\ts b }]\left(\frac{n}{2} +\frac{1}{N} \sum_{i<j \le a\text{ or }a<i<j} ( i\hspace{0,1cm} j) -\frac{1}{N}\sum_{i\le a<j} \langle i \hspace{0,1cm} j \rangle \right).$$
\end{lemma}

For any permutation $\sigma\in\Sy_n$, $\varphi_t(\sigma)= N^{-\# \sigma}\Tr(\sigma    \esp[U_t^{\ts n}])$   and the Lemma \ref{lemEDPpol} reduces to  this more general one. 


\begin{dem} We shall use the stochastic differential equation (\ref{EDSU}) and apply It� formula. First, writing the $\uN$-valued Brownian motion $(K_t)_{t\ge 0}$  as a sum of  independent real standard Brownian motions yields  that $$\la\!\langle dK_t \ts dK_t\ra\!\rangle =- \frac{1}{N}\sum_{1\le r,s \le N }E_{r,s } \ts  E_{s,r} dt = \frac{1}{N}( 1 \hspace{0,1cm} 2) dt\in \End((\C^N)^{\ts 2})$$ 
and $$\la\!\langle dK_t \ts d\overline{K}_t \ra\!\rangle= \frac{1}{N}  \sum_{1\le r,s\le N}E_{r,s}\ts E_{r,s}= \frac{1}{N}\langle 1  \,2\rangle dt .$$
We can now use  the It� formula to get that the variational-bounded part of the variation of the semi-martingales $U_t^{\ts 2}$ and $U_t\ts \overline{U_t}$ are respectively $ U_t \ts U_t  .(-dt+\la\!\langle dK_t \ts dK_t\ra\!\rangle)=- U_t^{\ts 2}(1+ \frac{1}{N}( 1\hspace{0,1cm} 2)) dt$ and  $U_t\ts \overline{U_t}(-dt+ \la\!\langle dK_t \ts d\overline{K}_t \ra\!\rangle )= -U_t\ts \overline{U_t}(1-\frac{1}{N}\langle 1 \, 2\rangle) dt$.  The same analysis yields that the variational-bounded part of the variation of the semi-martingale $U_t^{\ts a}\ts  \overline{U}_t^{\ts b }$ is
 $$U_t^{\ts a}\ts  \overline{U}_t^{\ts b }\left(-\frac{n}{2} -\frac{1}{N} \sum_{i<j \le a,\text{ or }a<i<j} ( i\hspace{0,1cm} j) +\frac{1}{N}\sum_{i\le a<j} \langle i \hspace{0,1cm} j \rangle \right)dt.$$\end{dem}

\section{Free energy, words in unitary Brownian motions}


The convergence of the above  paragraph can be considered as a law of large numbers for the traces of words in  unitary Brownian motion.  We aim at studying their Laplace transform and at deriving from this study a central limit theorem. Note that if $(K_t)_{t\ge 0}$ is a $\uN$ Brownian motion as defined above, for any $t\ge 0,$  $N^{-2}\log\esp[e^{N\Tr(K_t)}]= N^{-2}\log\esp[e^{\la K_t, i\Id_N\ra}]=t$ and $N^{-2}\log\esp[e^{N\Tr(K_tK_t^*)}]=N^{-2}\log\esp[e^{-\|K_t\|^2}]=\log\esp [e^{-B_t^2}]$, where $B_t$ is the marginal of a standard real Brownian motion.  These two naive examples suggest that the scaling chosen in Theorem \ref{Theom Energy Words}  is the good one.  We shall prove it in the following by  estimating cumulants.


\subsection{Laplace transforms and cumulants of traces} 



%As the cumulants of random variables are multilinear, the notion of cumulants extends to tensor-valed random variables.
\subsubsection{Scaling of cumulants\label{Scaling cumulants}} For any bounded random variable $X$,  the function $\log \esp[e^{zX} ]$   is analytic on a neighborhood of $0$. We denote its analytic expansion 
$$\log \esp[e^{zX} ]=\sum_{n\ge 1}\frac{C_n(X)}{n!}z^n ,$$
the coefficients $(C_n(X))_{n\ge 1}$ are called the cumulants of the random variable $X$. 
 We are interested here  in the behavior in $N$ of $N^{-2}\log\esp[e^{N\Tr(A_N)}]$, hence of the rescaled cumulant $$N^{n-2}C_n(\Tr(A_N)),$$ where the $A_N$ are   bounded random matrices of $M_N(\C)$, uniformly bounded in norm.  
 
 \subsubsection{Cumulants}These coefficients are related to the moments of $X$ via a M�bius inversion formula. For any $n\in\N^*,$ the set $\Pc_n$ of partitions of $\{1,\ldots,n\}$  is endowed with a partial  order $\lec$, such that for $\pi,\nu\in\Pc_n$,  $\pi\lec \nu $ if the blocks of $\pi$ are included in the one of $\nu$. It has a maximum and a minimum that we denote respectively by $1_n$ and $0_n$. Each partition $\pi$ has $\#\pi$  blocks. For any sequence of complex numbers $(\alpha_A)_{A\subset \{1,\ldots ,n\}},$ let us set  for any  partition $\pi\in\Pc_n,$    $$\alpha_{\pi}=\prod_{A\in \pi} \alpha_A.$$  
Then, there exists a unique sequence $(\beta_{\pi,\nu}(\alpha))_{\pi\lec \nu}$ such that for any two  partitions $\pi\lec\nu$, 
$$\alpha_{\nu}=\sum_{\pi \lec\pi'\lec \nu} \beta_{\pi, \pi'}(\alpha). $$ 
For any $\pi,\nu\in \Pc_n$ with $\displaystyle \nu \lec \pi$ and $\pi\not=\nu$, we set $\beta_{\pi,\nu}(\alpha)=0.$
If $X_1\ldots, X_n$ are  bounded complex  random variables and for any $A\subset \{1,\ldots,n\}$, $\alpha_A= \esp[\prod_{i \in A} X_{i}]$, let us set  $C_n(X_1,\ldots,X_n)= \beta_{0_n,1_n}(\alpha)$ and similarly, for any  pair of partitions $\pi,\nu,$  $C_{\pi,\nu}(X_1,\ldots, X_n)= \beta_{\pi,\nu}(\alpha)$.  Then, the following expansion holds for any $z\in \C^n$  in a neighborhood of $0$, 
$$\log\esp[e^{z_1X_1+\ldots+z_nX_n}]=\sum_{\substack{k\ge 1\\1\le i_1,\ldots,i_k\le n}}C_k(X_{i_1},\ldots,X_{i_k})\frac{z_{i_1}\ldots z_{i_k}}{k!}.$$
If $Y$ and $Z$ are bounded random variables coupled with $X$ and  if $z\in \C$ is in a neighborhood of $0,$  then
$$\frac{\esp[Ye^{zX}]}{\esp[e^{zX}]}=\sum_{k\ge 0}\frac{C_{k+1}(Y,X_{i_1},\ldots,X_{i_k})}{k!} $$
and 
$$ \frac{\esp[Y Z e^{zX}]}{\esp[e^{zX}]}-\frac{\esp[Y e^{z X}]\esp[ Z e^{z X}]}{\esp[e^{zX}]^2}=\sum_{k\ge 0}\frac{C_{k+2}(Y,Z,X,\ldots,X)}{k!}. $$
The coefficient $C_n(X_1,\ldots,X_n)$  is called a \emph{cumulant} and is symmetric in the variables $X_1,\ldots ,X_n$.    The coefficients $(C_{\pi,\nu}(X_1,\ldots,X_n))_{\pi\lec\nu}$ are called \emph{relative cumulants}.  For any pair $\pi\lec \nu$ and $A\in \nu$,  let us denote by $A_\pi$ the set of blocks of $\pi$ included in $A$, then
$$C_{\pi,\nu}(X_1,\ldots,X_n)= \prod_{A\in \nu} C_{\#A_\pi}(\prod_{i\in B} X_{i}, B\subset A \text{ and } B\in \pi).$$ 


\subsubsection{Tensor valued cumulants}  Let us fix some notations for tensors. For any  finite dimensional vector space $V$ and any finite set $A$, let us denote by $V^{\ts A} $ the vector space of multilinear map on $\left({V^*}\right)^A$ and for any $n\in\N^*$ identify  $V^{\ts \{1,\ldots,n\}}$ with  $V^{\ts n}.$    Any function  $X: \{1,\ldots,n\}\to V$ defines an elementary element of $V^{\ts A}$ that we denote by $\bigotimes_{i\in A} X_i $.   Any partition $\pi \in \Pc_n $ defines  a multilinear map $\prod_{A\in\pi} V^{\ts A}  \to V^{\ts n}, (\alpha_A)_{A\in\pi}\mapsto \bigotimes_{A\in \pi}\alpha_A ,$ such that for any $X\in V^n$, 
$\bigotimes_{A\in \pi} \bigotimes_{i\in A} X_i = \bigotimes_{i \in \{1,\ldots, n\}} X_i.$  For instance, for $v_1,\ldots,v_4\in V$,  if $\alpha_{\{1,3\}}= v_1\ts v_2\in V^{\ts 2}\simeq V^{\ts\{1,3\}}$ and $\alpha_{\{2,3\}}=v_3\ts v_4\in V^{\ts 2}\simeq V^{\ts\{2,4\}}$,
$$\bigotimes_{A\in \{\{1,3\},\{2,4\}\}} \alpha_A= v_1\ts v_3\ts v_2\ts v_4.$$ 
For any sequence $(\alpha_A)_{A\subset \{1,\ldots,n\}}$ such that for any $A\subset \{1,\ldots,n\}$, $\alpha_A\in V^{\ts A}$, let us set for any partition $\pi\in\Pc_n$,
$$\alpha_\pi=\bigotimes_{A\in\pi}\alpha_A\in V^{\ts n}.$$ 
Then, there exists a unique sequence $(\beta_{\pi,\nu})_{\pi\lec \nu}$ such that for any two  partitions $\pi\lec\nu$, 
$$\alpha_{\nu}=\sum_{\pi \lec\pi'\lec \nu} \beta_{\pi, \pi'}(\alpha)\in V^{\ts n}. $$ 
If $X_1\ldots, X_n$ are  bounded random variables valued in $V$  on a probability space $(\Omega,\Bc,\Prob)$ and for any $A\subset \{1,\ldots,n\}$, $\alpha_A= \esp[\Bts_{i \in A} X_{i}]$, note that for any  pair of partitions $\pi, \nu,$  $C_{\pi,\nu}(X_1, \ldots,  X_n)=\beta_{\pi,\nu}(\alpha)$ is  $n$-linear as a function on the space $\Ld_\infty(\Omega,\Bc,\Prob)\ts V$.  We then define linear functions  on $\Ld_\infty(\Omega,\Bc )\ts V^{\ts n}$  by setting for any random variables  $X_1,\ldots,X_n\in \Ld(\Omega,\Bc,\Prob)\ts V$, $C_{\pi,\nu}(X_1\ts \ldots\ts  X_n)= C_{\pi,\nu}(X_1, \ldots\,  X_n)$ and  $C_n(X_1\ts \ldots\ts X_n)=C_{0_n,1_n}(X_1, \ldots, X_n)$. For example, if $A$ and $B$ are two bounded random vectors of $V,$
$$C_2(A\ts B)= \esp[A\ts B]-\esp[A]\ts \esp[B]\in V^{\ts 2},$$
$$C_{\{\{1,2\},\{3\}\},1_3}(A\ts B \ts C)= \esp[A\ts B\ts C]-\esp[A\ts B]\ts \esp[C]\in V^{\ts 3}.$$
If $A_1,\ldots, A_n$ are random matrices in $M_n(\C^N)$  with bounded operator norms    and $\pi,\nu\in\Pc_n$ is a pair of partitions,  then 
$$C_{\pi,\nu}(\Tr(A_1),\Tr(A_2),\ldots, \Tr(A_n))= \Tr_{(\C^N)^{\ts n}}(C_{\pi,\nu}(A_1\ts\ldots\ts A_n)). $$
If $\sigma\in\Sy_n $ is a permutation whose orbits are included in blocks of the partition $\nu$, then 
\begin{equation}\label{Tensor Cumulants and usual Cumulants}
C_{\#\nu}(\prod_{\substack{(i_1\cdots i_k)\text{ cycle of }\sigma\\\text{included in }B}}\Tr(A_{i_1}\cdots A_{i_k}), B\in \nu)=\Tr( C_{\nu, 1_n}( \sigma A_1 \ts A_2\ts\ldots\ts A_n)).
\end{equation}

\subsubsection{Cumulant of  exponential tensors} For any $n\in \N^*$, let us denote by $\Pk_n$ the set of subset of $\{1,\ldots,n\}.$ For any $A,B\in \Pk_n $, with $B\subset A$, let us write $\overline{B}^A$  for the smallest partition of $A$ for $\lec$, containing   $B$ as a block, and set for any endomorphism $T\in \End(V^{\ts B})$, $\overline{T}^A= \Bts_{S\in\{B,A\setminus B \}}\alpha_S \in \End(V^{\ts A}),$ where $\alpha_B= T$ and $\alpha_{A\setminus B}=\Id_{V^{\ts A\setminus B}}.$   If $A=\{1,\ldots,n\}$, we shall write $\overline{T}$ for $\overline{T}^A.$
\begin{lemma} \label{Lemma exp cumulants}Let $(T_A)_{A\in \Pk_n, \#A \ge 2}$ be a family of endomorphisms such that for any $A\in\Pk_n$, $T_A\in \End(V^{\ts A}).$ 

i) If for any $A\in \Pk_n$,   $$\alpha_A= \exp[\sum_{B\subset A, \#B\ge 2}\overline{T_B}^A] ,$$  then,   for any pair of partitions $\pi\lec\nu $ in $\Pc_n,$ 
$$\beta_{\pi,\nu}(\alpha)= \sum_{k\ge0 }\frac{1}{k!} \sum  \overline{T_{A_1}}\ldots \overline{T_{A_k}}\in \End(V^{\ts n}),$$
where the second sum is over sequences $A_1,\ldots, A_k\in\Pk_n$, with $ \#A_i\ge 2$ for any $1\le i\le k$ and $\pi\vee A_1\vee A_2\vee \ldots \vee A_k= \nu$.


ii) For any $\mu\in \Pc_n$, let us set for each  block $C\in \mu,$  $L_C=  \sum_{B\subset C,\#B\ge 2}\overline{T_B}^A.$ If for any $t\in \R^\mu$ and $A\in \Pk_n,$
$$\alpha_A(t)= \exp[\sum_{c\in \mu}t_C L_C ],$$
then,  for any pair of partitions $\pi,\nu \in\Pc_n,$  $\beta_{\pi,\nu}(\alpha(\cdot))$ is a differentiable map and for any $t\in \R^\mu$ and $C\in \mu,$
$$\frac{d}{dt_C}\beta_{\pi,\nu}(\alpha(t))= \sum_{B\subset C,\#B\ge 2} \beta_{\pi\vee B,\nu}(\overline{T}_B\alpha(t)).$$
\end{lemma}
\begin{proof}i) For any $\pi,\nu\in \Pc_n,$ let $\tilde{\beta}_{\pi,\eta}$ the right-hand-side  of the formula of the statement. For any $\nu\in \Pc_\nu,$ the expansion of tensors of exponential $\alpha_\nu(\alpha)$ equals to the sum $\sum_{k\ge0 }\frac{1}{k!} \sum  \overline{T_{A_1}}\ldots \overline{T_{A_k}},$ where the second sum is over sequences $A_1,\ldots, A_k\in\Pk_n$, with $ \#A_i\ge 2$ for any $1\le i\le k$ and $ A_1\vee A_2\vee \ldots \vee A_k\lec \nu.$ This expression is equal to $\sum_{\pi\lec\nu}\tilde{\beta}_{\pi,\nu}$. 

ii) For any  $t\in \R^\mu$ $\pi,\nu\in \Pc_n, $ and $B\in \Bc_n,$ with $\#B\ge 2,$ $\beta_{\pi\vee B,\nu}(\overline{T}_B\alpha(t))=\overline{T}_B\beta_{\pi\vee B,\nu}(\alpha(t)).$  Note that the familly of endomorphisms $(L_C)_{C\in\mu}$ is commutative. For any $C\in\mu, $ differentiating with respect to $t_C$ the expansion of  $\beta_{\pi,\nu}(\alpha(t))$ given in  i)  yields the announced formula.
\end{proof}
   
   
  

\subsection{Words in independant unitary Brownian motions and their traces} 

We obtain here a differential system for the normalized cumulants in traces of   words of unitary brownian motions and show that the latter converge as $N\to\infty.$

\subsubsection{Partitioned and partial words\label{Section Partial words}}
For each positive integer $q$,  $W_q$  denotes  the monoid of words   in the alphabet made of  $2q$ symbols $x_1,\ldots,x_q ,x_1^{-1}, 
\ldots, x_q^{-1}$.    An element $w$ of $W_q$ writes down uniquely $x_{i_1}^{\epsilon_1}\ldots x_{i_n}^{\epsilon_n}$, with $
\epsilon_1,\ldots,\ep_n\in \{-1,1\}$. We call $n$  the length of $w$ and denote it by $\ell(w)$.  Its $p^{\text{th}}$ letter $x_{i_p}^
{\epsilon_p}$ is denoted by  $X_p(w)$ and for any $k\in \{1,\ldots,q\}$, we set $n^{\pm}_w(k) =\#\{r\in\{1,\ldots, n\}: X_r(w)=x_{k}
^{\pm}  \}$ and $\bar{n}_w(k)=n_w^+(k)+n_w^-(k).$    We call \emph{partitioned word} every couple $(S, \pi)$, where  $S$ is a tuple $
(w_1,\ldots,w_m)$ of   words  of $W_q$  and $\pi \in\Pc_m$.  Given such a couple, we set  $w(S)=w_1w_2\ldots w_m$ and $\ell(S,\pi)=
\ell
(w(S,\pi))$. We denote the set of partitioned words by $\Pc W_q$.  The symmetric group $\Sy_m$ acts diagonally on $\{(S,\pi): S\in W_q^m,
\pi\in \Pc_m\}$. A   
\emph{partial word} $[S,\pi]$ is the orbit  of a partitioned word $(S,\pi)\in\Pc W_q$, with $S\in W_q$, under the diagonal action of $
\Sy_m$. We denote the  set of partial 
words by $\overline{\Pc W_q}$. 


%A \emph{cyclic word} is the set $[w]$ of cyclic permutations  of a word $w\in W_q$. For pair of words $a,b\in W_q$, we set $a\sim_R b$ if and only if $a$ can be obtained from $b$ by inserting or deleting words of the form $x_i x_i^{-1}$ or $x_i^{-1}x_i$.  For any word $w\in W_q$, its equivalence classes for $\sim_R$ and   for the equivalence relation generated by $\sim_R $ and cyclic permutations admit both a  unique element of minimal length, called  respectively \emph{reduced} and \emph{cyclicly reduced}, that we denote   by $R(w)$ and  $CR(w)$. 
\subsubsection{Operations on  partitioned  words\label{Section CutJoin}}   For    any partitioned word $(S,\pi)\in \Pc W_q$ with $S=(w_1,\ldots, w_m)$ and $w=w(S,\pi)$, let us introduce two transformations of  $(S,\pi)$. For any pair  of positive integers $i,j$ such that  $1\le i<j \le \ell(S,\pi)$ and  $X_i(w)=X_j(w)^{\pm}=a$,  let us define  $\Tc^{\pm}_{i,j}(S,\pi)= (\Tc^{\pm}_{i,j}(S),\pi')\in \Pc W_q$  according to two cases. 

\vspace{0.2 cm}

\noindent 1. If the  $i^{\text{th}}$ and $j^{\text{th}}$  letters of $w$ belongs to the same word $w_k=  \lambda X_i(w)\mu X_j(w) \nu $,  then let us  set  $$\Tc^{+}_{i,j}(S)=(w_1,\ldots, w_{k-1}, \lambda a \nu,a \mu ,w_{k+1},\ldots,w_m),$$
if $X_i(w)=X_j(w)$,  
$$\Tc^{-}_{i,j}(S)=(w_1,\ldots, w_{k-1}, \lambda \nu,a\mu a^{-1} ,w_{k+1},\ldots,w_m),$$
if $X_i(w)=X_j(w)^{-1}$
and  in both cases $\pi'\in \Pc_{m+1}$ the partition  obtained from $\pi $ by substituting $l $ with $l+1$ for $l>k$ and adding $k+1$ to any block of $\pi$ including $k$. 

\vspace{0.1 cm}

\noindent 2. If the  $i^{\text{th}}$ and $j^{\text{th}}$  letters of $w$ belongs to two words $w_p=  \lambda X_i(w)\mu$ and $w_q=\nu  X_j(w) \chi $,  then  let us set   
$$\Tc^{+}_{i,j}(S)=(w_1,\ldots, \hat{w_p},\lambda a\chi\nu a\mu,\ldots,\hat{w_{q}},\ldots,w_m),$$
if $X_i(w)=X_j(w)$,  
$$\Tc^{-}_{i,j}(S)=(w_1,\ldots, \hat{w_p},\lambda \chi\nu a^{-1}a\mu,\ldots,\hat{w_{q}},\ldots,w_m),$$
if $X_i(w)=X_j(w)^{-1}$ and in both cases we let $\pi'\in \Pc_{m-1}$ be the image of the restriction of  $\pi $ to $\{1,\ldots, m\}\setminus \{q\}$ by the increasing bijection $\{1,\ldots,m\}\setminus\{q\}\to \{1,\ldots,m-1\}.$ 

\vspace{0,2 cm}

Let us make two remarks. In the first case the number of blocks of the partitioned words are constant and the number of words is increased by $1$. In the second one, the number of words is decreased by $1$, whereas the number of blocks of $\#\pi'$  is equal to $\#\pi$, if $p$ and $q$ belongs to the same block of $\pi$ and $\#\pi-1$ otherwise.  For any $f\in \{1,\ldots,q\},$ we define the sets 
$$\Nc^\pm_{2,w}(f)=\{(i,j)\in \{1,\ldots, \ell(w)\}^2: i<j, X_i(f)=X_j^\pm(f)\in \{x_f,x_f^{-1}\}\},$$
$$\Nc^{0,\pm}_{2,S,\pi}(f)=\{(i,j)\in \Nc_{2,w}^\pm(f) : \#\pi'=\#\pi-1 \text{ or }\# S' =\# S+1 \},$$
$$\Nc^{2,\pm}_{2,S,\pi}(f)=\{(i,j)\in  \Nc_{2,w}^\pm(f): \#\pi'=\#\pi \text{ and }\# S' =\# S-1 \},$$
$$\Sc_{2,S,\pi}(f)=\{(i,j)\in \Nc^\pm_{2,w}(f): \#\pi'=\#\pi\}.$$
and $$\Nc_{2,w}=\bigcup_{f=1}^q\Nc_{2,w}^+(f)\cup\Nc_{2,w}^-(f).$$


\begin{figure}\label{Cut Operation}\centering \includegraphics[height=2in]{CutOp}\caption{Cut  tranformations}
\end{figure}
\begin{figure}\label{Join Operation}\centering \includegraphics[height=1.5 in]{JoinOp}\caption{Join  tranformations}
\end{figure}


\subsubsection{A differential system\label{Section differential system}} 
Let us fix $q$ independant and identically distributed $\Un(N)$-valued  Brownian motions $(U_{1,s})_{s\ge 0},\ldots,(U_{q,s})_{s\ge 0}.$ For any word decomposed as $w=x_{i_1}^{\epsilon_1}\ldots x_{i_n}^{\epsilon_n}$, with $
\epsilon_1,\ldots,\ep_n\in \{-1,1\}$  and $U_1,\ldots ,U_q\in \Un(N),$ let us set 
$$w(U_1,\ldots, U_q)=U_{i_1}^{\ep_1}\ldots U_{i_n}^{\ep_n}$$
and for any vector $t\in \R_+^q,$  
 $$w_t^N = w(U_{1,t_1},\ldots,U_{q,t_q}). $$
 For any partial word $[S,\nu]\in \overline{\Pc W_q},$ with $S=(w_1,\ldots,w_m),$ we shall consider  for any $t\in\R_+^+$, 
 $$K_t(S,\nu)=C_{\#\nu}( \prod_{i\in A} \Tr(w^N_{i,t}), A\in \nu).$$
 
 \begin{lemma}\label{EDP cut and join}For any $f\in \{1,\ldots,q\}$ and any partitioned word $(S,\nu)\in\Pc W_q,$ 
 $$\frac{d}{dt_f}K_t(S,\nu)= -\frac{\overline{n}_w(f)}{2}K_t(S,\nu)-\frac{1}{N}\left( \sum  K_t(\Tc^{+}_{i,j}(S,\nu)) -    \sum K_t(\Tc^{-}_{i,j}(S,\nu))\right),$$
 where the sums are  over $(i,j)$ belonging respectively to  $\Nc_{2,w(S)}^-(f)$  and $\Nc_{2,w(S)}^+(f)$.
\end{lemma} 

\begin{proof} Let us fix a tuple $(w_1,\ldots,w_m)$ of  words, set $w=w_1\ldots w_m=x_{i_1}^{\epsilon_1}\ldots x_{i_n}^{\epsilon_n}$, with $\epsilon_1,\ldots,\ep_n\in \{-1,1\}$ and $\iota:\{1,\ldots,n\}\to\{1,\ldots,m\}$ the map induced by the decomposition of $w$ into $w_1,\ldots, w_m$.    For any partition $\nu \in \Pc_m,$ we denote by  $\overline{\nu}^{0}\in \Pc_{n}$ the biggest partition such that $\iota(\overline{\nu}^0)=\nu$. For any $\sigma\in \Sy_n$, let us denote by $S_\sigma$ the tuple  of words of the form $x_{i_{c_1}}^{\ep_{c_1}}\ldots x_{i_{c_k}}^{\ep_{c_k}}$, with $ (c_1\ldots c_k) $   cycle of $\sigma $, ordered by the position of their first letter in $w.$  We denote  $W_q(w)=\{S_\sigma:\sigma\in \Sy_n\}$  and  set  for any  $S\in W_q(w),$   $\sigma_S$ the unique permutation such that $S_{\sigma_S}=S.$  
%.fix a word $w =\in W_q$ and define a function $F_w$ on $\Sy_n\times M_N(\C)^{\ts n}$, by setting for any  $A_1,\ldots,A_n\in M_N(\C)$ and any $\sigma\in\Pc_n$, 
%$$ F_w(\sigma, A_1\ts\ldots \ts A_n)=\Tr_{(\C^N)^{\ts n}}(\sigma  A_1\ts \ldots\ts A_n)=\prod_{(c_1\ldots c_k) \text{ cycle of }\sigma} \Tr( A_{c_1}\ldots A_{c_k}).$$
%For any pair of  integers $i,j$, with $\ep_i=1$ and $\ep_j=-1$ and any permutation $\sigma$, let us define another permutation $\Tc_{i,j}^{-}(\sigma)\in \Sy_n$ with $X_i(w)=X_j(w)$,  $(i\,j)\sigma_S= \sigma_{\Tc^+_{i,j}(S)}.$ Furthermore, let us denote for any pair of distinct integers $i,j$
\vspace{0,1 cm}



Let us denote $\theta: M_N(\C^N)\to M_N(\C^N), M \mapsto M^t$, set for any integer $i\le \ell(w),$ $\delta_i=\frac{1-\ep_i}{2},$ and define  for any pair of distinct integers $i,j$ two  operators $T^+_{i,j}$ and $T^-_{i,j}$, acting on  $\End((\C^N)^{\ts \{i,j\}}),$ by setting for any $M\in \End((\C^N)^{\ts \{i,j\}}),$
\begin{align}\label{Transposition and Contraction}
T^+_{i,j}(M)&=M( i\hspace{0,1cm} j),\\
T^-_{i,j}(M)&=\theta_i^{\delta_i}\circ\theta^{\delta_j}_j\left(\theta^{\delta_i}_i\circ\theta^{\delta_j}_j(M)\langle i \hspace{0,1cm} j\rangle \right).
\end{align}
%$$\Tr_{(\C^N)^{\ts n}}( \sigma_S  T_{i,j}^{\pm}\left(w^{\ts n}(U_1,\ldots ,U_q) \right))=\Tr_{(\C^N)^{\ts n}}( \sigma_{\Tc^{\pm}_{i,j}(S)}  w^{\ts n}(U_1,\ldots ,U_q) ).$$  
For any collection  of  words $S\in W_q(w)$,  any pair $1\le i<j\le n$ and any $U_1,\ldots,U_n\in \Un(N)$ with $U^{\ep_i}_i=U^{\ep_j}_j,$ 
\begin{equation*}
\Tr_{(\C^N)^{\ts n}}( \sigma_S  \overline{T}_{i,j}^{\pm}\left(U_1\ts \ldots \ts U_n \right))=\Tr_{(\C^N)^{\ts n}}( \sigma_{\Tc^{\pm}_{i,j}(S)}  U_1\ts\ldots \ts U_n).
\end{equation*}
It follows that if  $U_1,\ldots, U_n $ are $\Un(N)$-valued random variables  with $U_i^{\ep_i}=U_j^{\ep_j}$, then for any partition $\pi\in\Pc_n $  such that $i$ and $j$ belongs to the same block of $\pi,$ 
%$$C_{\pi,1_n}\left(T_{i,j}^{\pm}\left(U_1\ts \ldots \ts U_n \right)\right)=T_{i,j}^{\pm}\left(C_{\pi,1_n}\left(U_1\ts \ldots \ts U_n \right)\right)$$
\begin{equation}\label{Cumulant relatif cut}
\Tr_{(\C^N)^{\ts n}}( \sigma_S  C_{\pi,1_n}\left(T_{i,j}^{\pm}\left(U_1\ts \ldots \ts U_n \right)\right))=\Tr_{(\C^N)^{\ts n}}( \sigma_{\Tc^{\pm}_{i,j}(S)}  C_{\pi,1_n}\left(U_1\ts \ldots \ts U_n \right)).
\end{equation}
 We shall now apply  Lemma \ref{EDPTens}  to the  tensors
$w_t^{\ts A} = \Bts_{k\in A} U_{i_k,t_{i_k}}^{\epsilon_k},$ for any $A\subset \{1,\ldots,n\},$ that we will simply denote by  $w_t^{\ts},$ for $A=\{1,\ldots,n\}.$ Setting $\overline{n}_{w,A}(f)=\#\{ i\in A: X_i(w)\in \{x_f,x_f^{-1}\} \}$,  Lemma \ref{EDPTens}  yields that for any $f\in\{1,\ldots, q\}$,
\begin{equation}
\frac{d}{dt_f}\esp[w_t^{\ts A}]= -\frac{\overline{n}_{w,A}(f)}{2}\esp[w_t^{\ts A}]- \frac{1}{N}\left(\sum  \overline{T^+_{i,j}}^A-\sum \overline{T^-_{i,j}}^A\right)\left(\esp[w_t^{\ts A}]\right),
\end{equation}
where the  first and the second sums are  over pairs $(i,j)\in A^2$ belonging respectively to $\Nc^+_{2,w}(f)$ and  $\Nc^-_{2,w}(f)$.
Consider now  a partitioned word $(S,\nu)$  with $S\in W_q(w)$.   According to (\ref{Tensor Cumulants and usual Cumulants}),
$$K_t(S,\nu)= \Tr_{(\C^N)^{\ts n} } ( C_{\overline{\nu}^0, 1_n}[ \sigma_{S} w_t^{\ts }]).$$
For any subset $A\subset \{1,\ldots, n\}$, with $\#A \ge 2$, let us set $T_A=0,$ if $\#A\ge 3 $ and 
\begin{equation} \label{transposition contraction signed}
T_A=\mp T^{\pm}_{a,b},
\end{equation}
if $A=\{i,j\},$ with $(a,b)\in \Nc_{2,w}.$ 
 Applying  Lemma \ref{Lemma exp cumulants} with the vector space $V=M_N(\C)$,  the familly of tensors $\alpha_A=e^{\frac{1}
 {2}\sum_{1\le f\le q} t_f \overline{n}_{w,A}(f)}\esp[w_t^{\ts A}], $ for $A\subset \{1,\ldots,n\},$  the partition $\{\Nc_{2,w}^+(f)
 \cup \Nc_{2,w}^-(f):1\le f\le q\}\in \Pc_n$  and considering the evaluation against $\Id_{(\C^N)^{\ts n}}$ yield the two following equalities. For any partition $\pi\in \Pc_n,$  
\begin{equation}\label{Cumulant words Tensor}
C_{\pi,\overline{\nu}^{0}}[ w_t^{\ts }]=e^{-\frac{1}{2}\sum_{1\le f\le q} t_f \overline{n}_w(f)}\sum_{k\ge 0}\frac{1}{k!N^k}
\sum t_{i_{a_1}}\ldots t_{i_{a_k}} \overline{T}_{\{a_1,b_1\}}\ldots \overline{T}_{\{a_k,b_k\}}(\Id_{(\C^N)^{\ts n}}),
\end{equation}
 where the second sum is over the sequences $(a_l,b_l)_{1\le l\le k}\in \Nc_{2,w}^k$  and $\pi\vee\overline{\{a_1,b_1\}}\vee \ldots\vee \overline{\{a_k,b_k\}}=\overline{\nu}^{0}.$ Moreover, for any $f\in \{1,\ldots,q\},$
   \begin{align*}\tag{*}\label{EDS Cumulant Tens}
\frac{d}{dt_f}\left(e^{\frac{1}{2}\sum_{1\le f\le q} t_f \overline{n}_w(f)}C_{\pi,1_n}(w_t^{\ts }) \right) = \sum_{(a,b)\in \Nc^+_{2,w}(f)\cup\Nc^-_{2,w}(f)} C_{\pi\vee \overline{\{a,b\}},1_n} \left(\overline{T}_{\{a,b\}}\left(w_t^{\ts }\right)\right).
\end{align*}
For any  pair $(a,b)\in\Nc_{2,w}$, if $p,q\in\{1,\ldots,m\}$ are such that the $a^{\text{th}}$ and the $b^{\text{th}}$ letters of $w$ belong respectively to $w_p$ and $w_q$, then  according to (\ref{Cumulant relatif cut}), 
\begin{align*}
\Tr_{(\C^N)^{\ts n}}(\sigma_S C_{\overline{\nu}^{0}\vee \overline{\{a,b\}},1_n}\left(  \overline{T}_{\{a,b\}}\left(w_t^{\ts}\right)\right))&=-\ep_a\ep_b\Tr_{(\C^N)^{\ts n}}(\sigma_{\Tc_{a,b}^{\ep_i\ep_j}(S)} C_{\overline{\nu}^{0}\vee \overline{\{a,b\}},1_n}\left(  w_t^{\ts}\right))\\
& =-\ep_a\ep_bK_t(\Tc^{\ep_i\ep_j}_{a,b}(S,\nu \vee \overline{\{p,q\}})).\label{cut tensor cut cycle}\tag{**}
\end{align*}
The  two equations  (\ref{EDS Cumulant Tens}) and (\ref{cut tensor cut cycle})  then implie the announced formula. \end{proof}




\subsubsection{Scaling and asymptotic expansion of the cumulants} Let us now introduce a scaling of the above functions that matches  the one of section \ref{Scaling cumulants} and that yields a differential system with initial conditions independant of $N$. For any partial word  $[S,\nu]\in \overline{\Pc W_q},$ with $S=(w_1,\ldots,w_m),$  the following quantity  
$$\vp_{t,N}([S,\nu])= N^{2(\#\nu-1)-n}C_{\#\nu}( \prod_{i\in A} \Tr(w_{i,t}), A\in \nu)$$
satisfies these two conditions. Let us define two operators on $\C^{\overline{\Pc W_q}}$, by setting for any function $\vp\in \C^{\overline{\Pc W_q}}$ and $(S,\nu)\in \Pc W_q,$ $L_f(\vp)([S,\nu])$ to be equal to
\begin{align*}
 -\frac{\overline{n}_{w(S)}(f)}{2}\vp([S,\nu])&+\sum_{(i,j)\in \Nc^{0,-}_{2,S,\pi}} \vp([\Tc_{i,j}^{-}(S,\nu)])-\sum_{(i,j)\in \Nc^{0,+}_{2,S,\pi}}  \vp([\Tc_{i,j}^{+}(S,\nu)] )
\end{align*}
 and 
  $$D_f(\vp)([S,\nu])= \sum_{(i,j)\in \Nc^{2,-}_{2,S,\nu}} \vp([\Tc_{i,j}^{-}(S,\nu)])-\sum_{(i,j)\in \Nc^{2,+}_{2,S,\nu}}  \vp([\Tc_{i,j}^{+}(S,\nu)]).$$

\begin{prop}\label{PropoSystDiff Cut and Join}For any $t\in \R_+^q,$  $N\in \N^*$ and $(S,\pi)\in \Pc W_q$,
$$\frac{d}{dt_f} \vp_{t,N}([S,\pi]) =  (L_f+\frac{1}{N^2}D_f).\vp_{t,N}([S,\pi])$$
and  $\vp_{0,N}([S,\pi])=1,$ if $\pi$ has one block and $0$ otherwise. As $N\to \infty,$ the sequence of functions $\vp_{t,N}$ converges pointwise  towards the unique function $\vp_t,$ such that for any $t\in \R_+^q$ and $ (S,\pi)\in \Pc W_q,$ 
$$\frac{d}{dt_f} \vp_{t}([S,\pi]) =  L_f.\vp_{t}([S,\pi])$$
and $\vp_{0}=\vp_{0,1}.$
\end{prop}
 For any $w\in W_q$, $t\in \R_+^q$ and $N\in \N^*$, we shall use the simpler notation  $\vp_{t,N}(w)= \vp_{t,N}([(w),0_1])$ and $\vp_{t,N}(w)= \vp_t([(w),0_1]).$
\begin{proof}It is a direct consequence of Lemma \ref{EDP cut and join}.
\end{proof}
\begin{corol} there exists a sequence of functions $(\psi_{t,g})_{g\ge 1}$ on $\Pc W_q$ such for any $(S,\pi)$,  the power series with coefficients  $(\psi_{t,g}([S,\pi]))_{g\ge 1}$ has a positive radius of convergence and for $N$ large enough,
$$\vp_{t,N}([S,\pi])=\vp_{t}([S,\pi])+\sum_{g\ge 1}N^{-2g}\psi_{t,g}([S,\pi]).$$
%For any words $w_1,\ldots,w_m\in W_q,$ $t\in \R_+^q,$  $N^{m-2}C_{m}(\Tr(w^N_{1,t}),\ldots,\Tr(w^N_{m,t}))$ converges as $N\to\infty$ towards $\vp_t((w_1,\ldots, w_m), 1_m)$ and $C_{m}(\Tr(w^N_{1,t}),\ldots,\Tr(w^N_{m,t}))- N^2\vp_t((w_1,\ldots,w_m),1_m)$ admits a limit,  as $N\to \infty,$ denoted by $\psi_t(w_1,\ldots, w_m)$.
\end{corol}
\begin{proof} For any fixed $n\in\N^*$, the operators $L_f,D_f$ preserve the finite dimensional space of functions supported on $\{x \in \overline{\Pc W}_q: \ell(x)\le n\}.$ The above expansion follows then easily from Proposition \ref{PropoSystDiff Cut and Join}.
\end{proof}
In particular, for any $n_1,\ldots,n_m\in \Z^*,$  $N^{m-2}C_m(\Tr(U_t^{n_1}),\ldots,\Tr(U_t^{n_m}))$ admits a limit as $N\to\infty.$  Together with the property of independance, stationarity  and invariance by unitary adjunction of multiplicative increments of the process $(U_t)_{t\ge 0}$ and   the notion of higher order freeness developed in \cite{HOPS}, this fact alone implies that for any $(S,\pi)\in \Pc W_q,$  $\vp_{t,N}([S,\pi])$ admits a limit as $N\to\infty.$  Nonetheless, this result does not give  easily an  expansion  in $N$. %\footnote{Though, it is not written in \cite{HOPS}, using Weingarten calculus, it should hold that knowing the asymptotic expansion in $N$ of two higher-order non commutative distribution  of two random  sub-algebra of $M_N(\C)$ that are invariant} 




\section{Two estimates on the cumulants} 


The proof of Theorem \ref{Theom Energy Words} relies on  two estimates on the above cumulants. The first bound gives a bound of their modulus that allows to extend them to broader class of words as we will see in section \ref{Yang-Mills}.   It is nonetheless too loose to  obtain a positive radius of convergence as stated in Theorem \ref{Theom Energy Words}. The second type of estimates gives a much sharper bound that allows to conclude.

\vspace{0,5 cm}


\subsection{All-order bounds}  For any  $t\in \R_+^q$,  let us define a scalar product $\la\cdot,\cdot\ra_t$ on $\R^m$, by setting for any $a,b\in \R^m,$
$$\la a, b\ra_t= \sum_{f=1}^q a_fb_f t_t.$$
For any word $w\in W_q$, we set $$A_t(w)=\la \overline{n}_w,\overline{n}_w\ra_t$$
and   for any $(S,\pi)\in \Pc W_q$ and $N\in \N^*$,  we define inductively a sequence by setting 
$\psi_{t,0,N}([S,\pi])= \vp_{t,N}([S,\pi])$ and for any $g\in\N,$
$$\psi_{t,g+1,N}([S,\pi])=N^2\left(\psi_{t,g,N}([S,\pi])-\psi_{t,g}([S,\pi])\right).$$
\begin{lemma}  \label{Exponential Bound Amperean area}For any  words $w_1,\ldots,w_m\in W_q$,  $N\in\N^*,$ $t\in \R_+^q$, $
\lambda\in [0,1]$  and  $k\in \N,$
$$|\frac{d^k}{d\lambda^k} \vp_{\lambda t,N}([(w_1,\ldots,w_m),0_m])| \le  m^k(A_t(w_1)+ \cdots + A_t(w_m))^ke^{m(A_t(w_1)+\cdots +A_t(w_m))} $$
and 
$$ |\psi_{ t,k,N}([(w_1,\ldots,w_m),0_m])|\le   m^k(A_t(w_1)+ \cdots + A_t(w_m))^{k}e^{m (A_t(w_1)+\cdots +A_t
(w_m))}.$$
\end{lemma}
\begin{proof}For any $p\in\N$ and $M\in M_p(\C)$, let us set  $\|M\|= \max_{i\in\{1,\ldots, p\}}\sum_{j=1}^p|M_{i,j}|.$ Recall 
that $\|\cdot\|$ is a sub-multiplicative norm on $M_p(\C)$, such that for any matrix $M\in M_p(\C)$ and $v\in \C^p$, $\displaystyle 
\max_{i\in \{1,\ldots, p\}}| (Mv)_i| \le \|M\|\max_{i\in \{1,\ldots, p\}}|v_i| .$  Let us fix  a word $w\in W_q$ and denote by $B_
{w}$ the set of partial words $[S,\pi]\in \overline{\Pc W_q}$ with $n^{\pm}_{w(S)}=n^{\pm}_w$.   Note that $B_w$ is stable by the 
operations defined in section \ref{Section CutJoin}, so that for any  $f\in \{1,\ldots , q\}$,  the two operators $L_f$ and $D_f$ preserve  
the finite dimensional space $\Fc_{w}$ of functions on $\overline{\Pc W_q}$ with support in $B_w.$ For any $F\in \End(\Fc_w),$ let us 
set $$\|F\|=\max_{y\in B_w}\sum_{x\in B_w} F(\delta_{x})(y).$$
For any $[S',\pi']\in B_w$, there are at most $\frac{\bar{n}_{w}(f)(\bar{n}_{w}(f)-1)}{2}$ elements $[S,\pi]$ of $B_w$ such that $(S',\pi')
$ is obtained from $(S,\pi)$ by a transformation of the form $\Tc_{i,j}^{\pm}$ with $(i,j)\in \Nc_{2,w(S)}$. It follows that the restriction of 
$L_f$ and $D_f$ to $\Fc_w$ satisfy the following inequality
  $$\max\{\|{L_f}_{|\Fc_w}\|, \|{D_f}_{|\Fc_w}\|,\|{L_f}_{|\Fc_w}+\frac{1}{N^2}{D_f}_{|\Fc_w}\|\}\le \frac{\overline{n}
  _w(f)^2}{2}.$$
Let us set  $L=\sum_{f=1}^qt_f{L_f}_{|\Fc_w}$ and $D=\sum_{f=1}^qt_f{D_f}_{|\Fc_w}$. Then, according to Proposition \ref
{PropoSystDiff Cut and Join}, 
  $${\vp_{t,N}}_{|{B_w}}=e^{L+\frac{1}{N^2}D}(\vp_0)_{|{B_w}}$$
and 
$${\vp_t}_{|{B_w}}=e^{L}(\vp_0)_{|{B_w}}.$$
Let us recall that for any matrices $A,B\in M_p(\C),$ \begin{equation}
e^{A+B}-e^A=\int_0^1 e^{s (A+B)}B e^{(1-s) A} ds. \label{Duhamel}
\end{equation}
Using iteratively this formula  together with the former two equations yields that for any $g\ge 1,$
\begin{equation}
{\psi_{t,g,N}}_{|B_w}= \int_{0<s_1<s_2<\ldots<s_g<1}e^{s_1(L+\frac{1}{N^2}D)}De^{(s_2-s_1)L}D\ldots D e^{(1-s_g)L}ds 
({\vp_0}_{|B_w}).
\end{equation}
Therefore, 
\begin{align*}
\max_{[S,\pi]\in B_w}\{g!{|\psi_{t,g,N}}([S,\pi])| \}&\le  \|D\|^g e^{\max\{\|L+\frac{1}{N^2}D\|,\|L\|\}}\max_{[S,\pi]\in 
B_w}\{\vp_0([S,\pi]) \}\\
&\le 2^{-g}(\sum_{f=1}^q t_f \overline{n}^2_w)^g e^{\frac{1}{2}\sum_{f=1}^q t_f \overline{n}^2_w}=2^{-g}A_t(w)^ge^{\frac{A_t
(w)}{2}}.
\end{align*}
Besides, for all $\lambda\in [0,1]$  and  $r\in \N,$
$$\frac{d^r}{d\lambda^r} {\vp_{\lambda t,N}}_{|B_w}= (L+\frac{1}{N^2}D)^re^{\lambda(L+\frac{1}{N^2}D)}({\vp_0})_{|B_w},
$$
so that  the left-hand-side is uniformly bounded by $2^{-r}A_t(w)^re^{\frac{A_t(w)}{2}}$ on $B_w$.   Now for any  words $w_1\ldots w_m
\in W_q$ with $w_1\ldots w_m=w,$ $\overline{n}_w=\overline{n}_{w_1}+\ldots +\overline{n}_{w_m} $ and we simply use the bound 
$A_t(w)\le (2m-1) (A_t(w_1)+\ldots +A_t(w_m))$  to conclude.\end{proof}
Note that these first estimates are very loose. For example,  for $w=x_1^n,$ the lemma shows that $|\vp_t(w,0_1)|\le e^{t_1n^2},$ when we
 have\footnote{The right decay of this moment sequence is  a least polynomial, as the measure $\mu_{t_1}$ is absolutly continuous with respect to 
 the Lebesgue measure with a density that is H�lder continuous. The regularity of the density does depend on $t_1$, there are three regimes:  
 $t_1<4,$ $t_1=4$ and $t_1>4$, see remark 6.8. of \cite{ThierrySW}.  } the simple bound $|\vp_t(w,0_1)|= |\int_\U \omega^n \mu_
 {t_1}(d\omega)|\le 1 $. Furthermore, for this same word, it yields for any $m\in\N^*$, $|N^{m-2}C_m(\Tr(U_{t_1}^n))|\le e^{t_1 
 n^2m^2}$ and the exponential power series of the sequence on the right-hand-side diverges. 
 
 
 \subsection{Sharper bounds for the first and second orders} For any  positive integer $m,$ let us denote by $
 \Cc_m$  the set of Cayley trees on $m$ vertices. For any $w_1,\ldots,w_m\in W_q$, $T\in \Cc_m$,   we set $$T_t(w_1,\ldots,w_m)=  
 \prod_{\{i,j\} \text{ edge of }T}  \la \overline{n}_{w_i}, \overline{n}_{w_j}\ra_t  $$
 and $$\tilde{T}_t(w_1,\ldots,w_m)= \prod_{\{i,j\} \text{ edge of }T}  \la  n_{w_i}, n_{w_j}\ra_t,  $$
if $m\ge 2,$ and $T_t(w_1)=\tilde{T}_t(w_1)=1$,  otherwise.

\begin{prop} For any   words $w_1,\ldots,w_m\in W_q$ and any $N\in\N^*,$
$$|\vp_{t,N}([(w_1,\ldots, w_m),0_m])|\le  \sum_{T\in \Cc_m} T_t(w_1,\ldots, w_m) $$\label{Cumulant's control}
and 
$$ \vp_{ t,N}([(w_1,\ldots, w_m),0_m])=\sum_{T\in \Cc_m}\tilde{T}_t(w_1,\ldots,w_m)+R_{t,N}(w_1,\ldots,w_m),$$
with $|R_{t,N}(w_1,\ldots,w_m)|\le m^m(A_t(w_1)+ \cdots + A_t(w_m))^{m}e^{m (A_t(w_1)+\cdots +A_t
(w_m))}.$
\end{prop}
The second estimate shows that it is optimal for tuples of words $w\in W_q,$ such that  $\overline{n}_w= n_w.$  
 
\begin{proof}  Let us consider $q$ independant Brownian motions $(U_{1,t})_{t\ge 0},\ldots,(U_{q,t})_{t\ge 0}$, on $\Un(N),$ fix a tuple  of 
$m$ words $S_0=(w_1,\ldots,w_m)\in W^m_q$, and set  $w=w_1\ldots w_m=x^{\ep_1}_{i_1}\ldots x^{\ep_n}_{i_n}$,  $w_t^\ts=U_{i_1,t_{i_1}}^{\ep_1}\ts\cdots\ts U_{i_n,t_{i_n}}^{\ep_n}$, $\sigma_0=(1\ldots \ell(w_1))\times \cdots \times (1\ldots \ell(w_m)) $  and $ \pi_0\in\Pc_n$ the set of its orbits ordered by their first element.  Let us further use the same notations as in the first paragraph of proof of Lemma \ref{EDP cut and join}.   According our choice of scaling and (\ref{Tensor Cumulants and usual Cumulants}), 
$$\vp_{t,N}([(w_1,\ldots, w_m),0_m])=N^{m-2} \Tr_{(\C^N)^{\ts n}}\left( \sigma_0 C_{\pi_0,1_m}(w_t^{\ts })\right).$$
The argument goes as follows: we first give an expression of the right-hand-side as an integral over the standard simplex as we did in the proof of Lemma \ref{Exponential Bound Amperean area} but using a different decomposition of the operator $\sum_{f=1}^q t_f(L_f+N^{-2}D_f).$ Then, we show that the integrands are normalized traces of unitary matrices.

\vspace{0,1 cm}

\noindent\textbf{Step 1:}Let us remind the definition of $\overline{T_{a,b}}$ given in  (\ref{Transposition and Contraction}) and (\ref{transposition contraction 
signed})  and  that according to (\ref{Cumulant words Tensor}),
$$ e^{\frac{1}{2}\sum_{1\le f\le q} t_f \overline{n}_w(f)} C_{\pi,1_m}(w_t^{\ts })= \sum_{k\ge 0}\frac{1}{k!N^k}\sum t_{i_{p_1}}
\ldots t_{i_{p_k}} \overline{T_{p_1,q_1}}\ldots \overline{T_{p_k,q_k}}(\Id_{(\C^N)^{\ts n}}),$$
where the second sum is over the set $\Upsilon_k $ formed by sequences of ordered pairs $(p_l,q_l)_{1\le l\le k}\in \Nc_{2,w}^k$, 
with $\vee_{l=1}^k\overline{\{p_l,q_l\}}\vee \pi_0=1_n$. For any $m\in \N^*$, if   $\Delta^{m-1}$ denotes the standard simplex  $\{s\in \R_+^{m}: s_1+\ldots +s_m=1\}$ and $ds $ the Lebesgue measure on $\Delta^{m-1},$ then, for any matrices $A_1,\ldots, A_{m}\in M_p(\C)$, 

\begin{align}
&\sum_{k_1,\ldots,k_{m}\ge 0 }\frac{1}{(k_1+\ldots+ k_{m}+m-1)!}(A_1+\ldots+A_{m})^{k_{m}}A_{m} \ldots (A_1+A_2)^{k_1}A_2 A_1^{k_1}\label{Dumahelplus}\\
&=\int_{\Delta^{m-1}}e^{s_m(A_1+\ldots+A_m)}A_m\ldots e^{s_2(A_1+A_2)}A_1e^{s_1A_1}
ds.\nonumber
\end{align}
For every $k\ge 1,$ each sequence $\gamma=(p_l,q_l)_{1\le l\le k}\in \Upsilon_k $   is the 
interlacement of a sequence $\gamma\in\Upsilon_{m-1}$ with $m$ sequences $\gamma_1\in\Nc_{2,w}^{k_1},\ldots, \gamma_{m}\in 
\Nc_{2,w}^{k_m}$ with $k_1+\ldots+k_m+m-1=k$ and there is a bijection 
\begin{equation*}
\Upsilon_k\to \Upsilon_{m-1}\times\bigcup_{k_1+\ldots+k_m=k-m+1}\prod_{l=1}^m \Nc_{2,w}^{k_l}.\tag{*}
\label{decomposition of paths in the partition poset}
\end{equation*}
For any partition $\nu\in\Pc_m$,  let us set  $A_{\nu}=\frac{1}{N}\sum_{(a,b)\in\bigcup_{f=1}^q\Sc_{2,w,\nu}(f) }\overline{T}_
{a,b}.$  Let $\Psi_{m}$ be the 
set of strictly increasing sequences $(\nu_i)_{i=1}^m\in\Pc_m^m$, with $\nu_1=0_m$ and $\nu_m=1_m.$ Any sequence of $\Upsilon_{m-1}$ induces 
an element of $\Psi_m$, so that,  rewriting the second sum of  (\ref{Cumulant words Tensor}) thanks to the bijection (\ref{decomposition of paths in the 
partition poset}),  and then applying (\ref{Dumahelplus}), yields that  $e^{\frac{1}{2}\sum_{1\le f\le q} t_f \overline{n}_w(f)} C_{\pi,
1_m}(w_t^{\ts })$ equals to 
$$\sum_{(\nu_k)_k\in\Psi_m} \int_{\Delta^{m-1}} e^{s_mA_{\nu_m}}(A_{1_m}-A_{\nu_{m-1}})\ldots e^{s_2 A_{\nu_2}}(A_{\nu_2}-A_{\nu_1})e^{s_1A_{\nu_1}}(\Id_{(\C^N)^{\ts n}})ds. $$
\textbf{Step 2:} Now, for any partition $\nu\in\Pc_m,$ according to the very first equation on tensors (\ref{lemEDPpol}) that we obtained,   for any $v
  \in \End((\C^N)^{\ts n})$ and $s\ge 0,$
  $$ e^{-\frac{1}{2}\sum_{f=1}^q s  t_f  n_w(f) +s A_\nu }(v)= \ts_{A\in \overline{\nu}^{0}} \esp[\Bts_{k\in A} 
  U_{i_k,st_{i_k}}^{\ep_k}] v.  $$  
For any partition $\mu\in\Pc_m$, let us denote by $\underline{\mu}:\{1,\ldots,n\}\to \mu$ the belonging map and introduce $((U_{t,b,f})_{t\ge 0})_{b\in \mu, 1\le f\le q}$  a collection  of $q\#\mu $ independant 
$\Un(N)$-Brownian motions. Then, let us $(U^{\mu})_{\mu\in\Pc_n}$ be a collection of independant random variables such that for each $\mu\in \Pc_n$, 
 $$\left(U^\mu_{t_k,k}\right)_{t\in \R_+^n,  1\le k\le n}\overset{(law)}{=}\left(U_{t_k,\underline{\mu}(k),i_k}^{\ep_k}\right)_{t\in \R_+^n,  1\le k\le n}.$$  For any 
element $x\in \Psi_m$, let us denote by 
$\Upsilon(x)$ the set of sequences of pairs of integers in $\Upsilon_{m-1},$ inducing the sequence of partitions $x$.  For any $v\in 
\End((\C^N)^{\ts n}),$ let us write $L(v)$ for the endomorphism of left multiplication by $v$ on $\End((\C^N)^{\ts n}).$ It now follows 
from the last two formulas that $ N^{1-m} C_{\pi,1_m}(w_t^{\ts })$ equals to the image of $\Id_{(\C^N)^{\ts n}}$ by
$$ \sum_{\nu\in\Psi_{m}}\sum_{(p_k,q_k)_k\in\Upsilon(\nu)} \int_{\Delta^{m-1}}ds \esp\left[
L(\Bts_{k\in\{1,\ldots,n\}}{U^{\nu_{m}}_{s_{m} t_{i_k},k}}) \overline{T}_{p_k,q_k}\ldots \overline{T}_{p_1,q_1}
L({\Bts_{k\in\{1,\ldots,n\}}
 {U^{\nu_{1}}_{s_{1} t_{i_k},k}}})\right ].$$
Notice that for every matrices $U_1,\ldots,U_n\in \Un(N)$ and  any pair $(a,b)\in \Nc_{2,w}$ such that $U_a=U_b$,  $L(\Bts_{k\in 
\{1,\ldots,n\}} {U_{k}}) $ commutes with $\overline{T}_{a,b}.$  Therefore, if we set for any sequence $\nu\in \Psi_m,$  
$s\in \Delta^{m-1}$ and $1\le k\le n,$ 
 $$V_{s,k}^\nu=U_{s_{m}t_{i_k},k}^{\nu_m}\ldots U_{s_1t_{i_k},k}^{\nu_1}\in\Un(N),$$
   then,    $N^{1-m}\Tr_{(\C^N)^{\ts n}}\left( \sigma_0 C_{\pi,1_m}(w_t^{\ts })\right)$ equals to 
\begin{align*}
\sum_{\nu\in\Psi_{m}}\sum_{(p_k,q_k)_k\in
 \Upsilon(\nu)}\int_{\Delta^{m-1}}ds \esp[\Tr(\sigma_0 \overline{T}_{p_{m-1},q_
 {m-1}}\ldots \overline{T}_{p_1,q_1} ( \ts_{k\in\{1,\ldots,n\}}V^\nu_{s,k} ))].
\end{align*} 
For each  $\gamma=(p_k,q_k)_k\in \Upsilon_{m-1},$  if $w_\gamma$ denotes the word so that $\Tc^{\ep_{p_1}}_{p_1,q_1}\circ\ldots\circ\Tc^{\ep_{p_{m-1}}}_{p_{m-1},q_{m-1}}(S,\pi)=((w_\gamma),1_1)$ and $t_\gamma=\prod_{k=1}^{m-1} \ep_{i_
{p_k}}t_{i_{p_k}},  $ then for every tensor  $v\in\End((\C^N)^{\ts n}),$
   $$\Tr(\sigma_0 \overline{T}_{p_{m-1},q_
 {m-1}}\ldots \overline{T}_{p_1,q_1} (v))=t_\gamma  \Tr(\sigma_{w_\gamma}   v).$$
It remains now to unfold the above notations to get 
 \begin{equation}\label{Formula Cumulants one Trace}
\vp_{t,N}([(w_1,\ldots, w_m),0_m])=\sum_{\nu\in\Psi_{m}}\sum_{\gamma\in\Upsilon(\nu)}t_\gamma \int_{\Delta^{m-1}} \esp[
N^{-1}\Tr( w_\gamma\left(V^\nu_{s,1},\ldots,V^\nu_{s,n}\right) ))]ds.
\end{equation}
To conclude, note that the normalized trace in the integrand of the right-hand-side are bounded by $1$. Let $\Gamma:\Psi_m\to \Cc_m$  be the  
$(m-1)!$-to-one map  that sends every sequence $\nu= (\vee_{l=1}^{k}\overline{\{i_l,j_l\}})_{1\le k<m}$ to the Cayley tree with edges 
$(\{i_k,j_k\})_{1\le k< m}$.   Then, for every $\nu\in \Psi_m,$ $ \sum_{\gamma\in\Upsilon(\nu)}|t_\gamma|=\Gamma(\nu)
(w_1,\ldots,w_m)$ and the first bound of the statement follows.  Another consequence of (\ref{Formula Cumulants one Trace}), is that, in the Taylor expansion of $\vp_{\cdot,N}([(w_1,\ldots,w_m),0_m])$ around $0\in \R_+^q,$ the terms  of degree less than $m-2$  vanish, whereas, the sum of terms of degree $m-1$ is exactly $\sum_{\nu\in \Psi_m} \sum_{\gamma\in\Upsilon(\nu)}\frac{t_\gamma}{(m-1)!}. $ But, for any $T\in\Cc_m$ and $\nu \in \Gamma^{-1}(T),$ $\sum_{\gamma\in\Upsilon(\nu)}t_\gamma=\tilde{T}_t(w_1,\ldots,w_m).$ Hence, applying Proposition \ref{Exponential Bound Amperean area} yields the second estimate.
\end{proof}
\begin{remark} It follows from the definition that $$\vp_{t,1}([(w_1),0_1])=e^{-\frac{1}{2} \|n_{w_1}\|_t^2},$$
whereas the above proof  shows that  for any $m\ge 2,$
$$\vp_{t,1}([(w_1,\ldots, w_m),0_m])= \sum_{T\in \Cc_m} \tilde{T}(w_1,\ldots,w_m)e^{-\frac{1}{2}\|n_{w_1}+\ldots+n_{w_m}   \|^2_t}. $$\end{remark}
 For any $t\in\R_+,$ let us set $$\lambda_t=  (2t+1+2\sqrt{t(t+1)})e^{\sqrt{t(t+1)}-\frac{t}{2}}$$
 and for any word  $w\in W_q$ and $t\in \R_+^q,$ 
 $$\lambda_t(w)= \prod_{f=1}^q \lambda_{t_i}^{\overline{n}_w(f)}. $$
\begin{lemma}\label{Bound second order} For any   words $w_1,\ldots,w_m\in W_q$ with $m\ge 2$  and any $N\in\N^*,$
$$|\psi_{t,1,N}([(w_1,\ldots, w_m),0_m])|\le 2^{m-1}m^2\prod_{i=1}^m\lambda_{t}(w_i) \max_{1\le i\le m}{\| \overline{n}_{w_i}\|_t ^2} \sum_{T\in \Cc_m} T(w_1,\ldots, w_m).  $$  
\end{lemma}
\begin{proof} Let us fix $S_0=(w_1,\ldots,w_m)\in W^m_q$ with $m\ge 2$, $w= w_1\ldots w_m$ and consider $B_0$ the set of partitioned word obtained from $x_0=[S_0,0_m]$ by  a sequence of cut and join transformations.  Each such  sequence, inducing a partitioned word $x=[S,\nu]\in\Pc W_q$, also induces recursively a partition of $\{1,\ldots, m\},$ with as many blocks as $\nu$.   This partition only depends on $x$,  we denote it by $ \eta_x\in \Pc_m$ and set for $\eta\in \Pc_m,$ $B_\eta=\{x\in B_0: \eta_x=\eta\}.$ For any $\eta\in\Pc_m$, we consider  $\Psi^\eta=\{(\vee_{i=1}^{k}\overline{\{p_i,q_i\}})_{0\le k\le m-\#\eta}\in\Pc_m^{m-\#\eta+1}: \vee_{i=1}^{k}\overline{\{p_i,q_i\}}=\eta\}$ and $\Psi_\eta=\{(\vee_{i=1}^{k}\overline{\{p_i,q_i\}}\vee \eta)_{0\le k\le \#\nu-1}\in\Pc_m^{\#\eta}: \vee_{i=1}^{k}\overline{\{p_i,q_i\}}\vee \eta=1_m\}.$  For  any $N\in\N^*,$ according to Proposition \ref{PropoSystDiff Cut and Join} and Duhamel's formula  (\ref{Duhamel}),
$$\psi_{t,1,N}([S_0,0_m])=-\int_0^1 e^{(1-s) L_t}D e^{s(L_t+N^{-2} D_t)} (\vp_0)(x_0)ds.  $$
For any $ t\in \R^q$ and $x,y\in B_0,$ let   us set 
$$ Q_{t}(x,y)=   e^{ L_t}D_t (\delta_{y})(x),$$
so that for any $t\in\R_+^q,$ 
\begin{align*}
-\psi_{t,1,N}([S_0,0_m])&=\sum_{y\in B_0}  \int_0^1  Q_{(1-s)t}(x_0,y)  e^{s(L+N^{-2} D)} (\vp_0)(y)ds\\
&=\sum_{\eta\in\Pc_m}\sum_{y\in B_\eta} \int_0^1 \vp_{st,N}(y) Q_{(1-s)t}(x_0,y)  ds.\tag{$\diamondsuit$}\label{Relative cumulants Bound}
\end{align*}
For any $t\in \R_+^q$ and any increasing sequence $\nu\in\Pc_m^l$,   induced by a sequence of pairs of integers $(p_i,q_i)_{1\le i \le l},$ let us set 
$$\nu_t(S_0)=\prod_{k=1}^l \la \overline{n}_{w_{p_k}}, \overline{n}_{w_{q_k}}\ra_t.$$
Let us fix  $\eta\in \Pc_m.$ For any $y\in B_{\eta}$,   a slight modification of the proof of Proposition \ref{Cumulant's control} yields that for any $t\in\R_+^q,$
\begin{equation*}
|\vp_{t,N}(y)| \le  \frac{1}{(\#\eta-1)!}  \sum_{\nu\in \Psi_{\eta}}\nu_t(S_0).
\end{equation*}
For any $\pi\in \Pc_m$ and any linear operator $A$ on $\C^{B_0},$ let us define another operator $A^{\pi}$ by setting for any $\vp\in \C^{B_0}$ and $x\in B_0,$  $A^{\pi}(\vp)(x)=\sum_{y \in B_{\pi}}\vp(y)A(\delta_y)(x).$  The same argument yields that    
$$  \sum_{y\in B_\eta}  |Q_{t}(y) |\le  \sum_{\substack{\nu\in \Psi^\eta \\ \overline{\{p,q\}}\lec \eta}} \la \overline{n}_{w_{p}},\overline{n}_{w_{q}}\ra_t \nu_t(S_0) \int_{\Delta^{\#\eta-1}} \sup_{x\in B_{\eta}} \sum_{y\in B_{\eta}}|  e^{s_{\#\eta}L^{\nu_{\#\eta}}_t+\ldots +s_{1}L^{\nu_{1}}_t}(\delta_{y})(x)|ds. $$
To conclude, we shall now expand the exponential in the right-hand-side and use triangular inequality. For each $f\in \{1,\ldots,q\},$ let us define an operator  $\tilde{L}_f$ on $\C^{B_0},$ by setting for all $\vp\in \C^{B_0}$ and $x=(S,\pi)\in B_0,$
 $$\tilde{L}_f(\vp)(x)=\sum_{(a,b)\in \Nc^{+,0}_{2,S,\pi}}\vp(\Tc^+_{a,b}(x))+\sum_{(a,b)\in \Nc^{-,0}_{2,S,\pi}}\vp(\Tc^-_{a,b}(x)).$$
For any $x\in B_0,$  $\nu\in \Psi^\eta$ and $s\in \Delta^{\#\eta-1},$ 
\begin{align*}
\sum_{y\in B_{\eta}}|  e^{s_{\#\eta}L^{\nu_{\#\eta}}_t+\ldots +s_{1}L^{\nu_{1}}_t}(\delta_{y})(x)|&\le e^{-\frac{1}{2}\sum_{f=1}^q\overline{n}_w(f)t_f}\sum_{y\in B_{\eta}}  e^{\sum_{f=1}^qt_f\tilde{L}_f }(\delta_{y})(x)\end{align*}
For any tuple $S$ of words in $W_q$, let us denote, for each $f\in\{1,\ldots, q\},$ by $w_f(S)\in W_q,$ the word obtained from $w(S)$ by deleting the letters $x_{f'}$ and $x_{f'}^{-1},$ for $f'\not=f.$ Recall that for any $x=[S,\pi]\in \overline{\Pc W_q}$,  $\vp_0(x)=1$, if $\#\pi=1,$  and $0$, otherwise.  Then,  for any $x=[S,\pi]\in B_\nu,$ 
$$\sum_{y\in B_{\eta}}  e^{\sum_{f=1}^qt_f\tilde{L}_f }(\delta_{y})(x)\le \prod_{f=1}^q   e^{\sum_{f=1}^qt_f\tilde{L}_f }(\vp_0)([(w_f(S)),1_{\overline{n}_w(f)}]).  $$
Let us denote by $$\Lambda_t(w)= e^{-\frac{1}{2}\sum_{f=1}^qt_f\overline{n}_w(f)}\sup_{[S,\pi]\in B_0} \prod_{f=1}^q  e^{\sum_{f=1}^qt_f\tilde{L}_f }(\vp_0)([(w_f(S)),1_{\overline{n}_w(f)}]). $$
Gathering (\ref{Relative cumulants Bound})  with the last four inequalities yields that 
\begin{align*}
| e^{ \sum_{f=1}^qt_f\overline{n}_w(f)}\psi_{t,1,N}([S_0,0_m])|&\le  \sum_{\substack{\eta \in \Pc_m\\\nu^1\in \Psi^\eta, \nu^2\in \Psi_\eta}}\frac{\nu^1_t(S_0)\nu^2_t(S_0)}{(\#\eta -1)! (m-\#\eta)!}\Lambda_t(w)\sup_{1\le i\le m}\|\overline{n}_{w_i}\|^2_t  \\
&\le \Lambda_t(w)\sup_{1\le i\le m}\|\overline{n}_{w_i}\|^2_t \sum_{\nu\in \Psi_{m-1}}\sum_{k=1}^{m} \frac{\nu_t(S_0)}{(k-1) ! (m-k)!} \\
&\le  2^{m-1}  \Lambda_t(w)\sup_{1\le i\le m}\|\overline{n}_{w_i}\|^2_t \sum_{T\in \Cc_m}T(S_0).
\end{align*}
Then, the following lemma  implies the annouced bound.
\end{proof}

\begin{lemma}  For any word $w\in W_q$,  $t\in \R_+^q,$ $$\Lambda_t(w)\le \lambda_t(w). $$
\end{lemma}
\begin{proof} For any $n\in \N^*,$ let $\Cc W_1$ be the set of finite tuple $S$ of words  in $W_1$, with $\ell(w(S))=n.$  For any $S\in \Cc W_1(n)$, $(a,b)\in \Nc^\pm_{2,w(S)},$ let us define  $\Tc^\mp_{a,b}((w))$ as in 1. of section \ref{Section CutJoin}.  For any $f\in\{1,\ldots,q\},$ we define a linear operator $\hat{L}$ on $\C^{\Cc W_q},$ by setting for any  $\vp \in \C^{\Cc W_q}$ and $S\in \Cc W_q, $
$$ \hat{L}(\vp)(S)=\sum_{(a,b)\in \Nc^{0,+}_{2,w(S),1_n}(1)\cup \Nc^{0,-}_{2,w(S),1_n}(1)}\left(\vp(\Tc_{a,b}^+(S) )+\vp(\Tc_{a,b}^-(S) )\right).$$
For any $n\in \N^*$, let us denote here by $\vp_0\in \C^{\Cc W_1}$ the constant function equal to $1$ and  set for any $s\in \R_+,$   $\rho_s(n)= e^{  s\hat{L}}(\vp_0)((x_1^n))$.  Notice that for any word $w\in W_1,$ with $\ell(w)=n,$ $e^{  s\hat{L}}(\vp_0)((w))=\rho_s(n).$  Therefore,   for any $t\in \R_+^q$ 
\begin{equation*}
\Lambda_t(w)\le e^{-\frac{1}{2}\sum_{f=1}^qt_f\overline{n}_w(f)} \prod_{f=1}^q\rho_{t_f}(\overline{n}_{w}(f)). \tag{*}\label{Upperbound Free Mult HBM}
\end{equation*}
According to the definition of the operator $\hat{L},$ the family of functions $(\rho_\cdot(n))_{n\ge 0  }$ satifies the following differential system: for any $n\in \N^*$ and $s\ge 0,$
$$\frac{d}{ds}\rho_s(n)=n(1+ \sum_{p=1}^n \rho_s(p)\rho_s(n-p)),$$
$\rho_0(n)=1.$  Let us define a formal power series by setting for any $z$,  $$ \rho_s(z)= 1+\sum_{n\ge 1}\rho_s(n) z^n.$$ 
Then, for any $s\ge 0,$ $$\frac{d}{ds}\rho_s(z)= \rho_s(z)+2z\rho_s(z) \frac{d}{dz} \rho_s(z). $$
According to the Lemma 13 of \cite{BianeSegalBarg},  $(\rho_s(n))_{n\ge 0}$ is the sequence of moments of a hermitian operator (therein, denoted by  $\Lambda_{s}\Lambda_{s}^*$) acting on a separable Hilbert space and,  according to Proposition 11 of the same article, with spectrum
$$[\lambda^-_s,\lambda_s^+]=[(2s+1-2\sqrt{s(1+s)})e^{-\sqrt{s(s+1)}},(2s+1+2\sqrt{s(s+1)})e^{\sqrt{s(s+1)}}].$$
It follows that for all $n\in\N^*,$ $\rho_s(n)\le {\lambda^+_s}^n $ and the result then follows from (\ref{Upperbound Free Mult HBM}).
\end{proof}

%{\Large Remark definition cumulants for $C^{m-1+\ep}$ functions. Conjecture growth cumulants }

\section{Applications}


\subsection{Asymptotics of the free energies\label{Section Free energies BM}}

For any function $V\in \C^{W_q}$, let us set  $\|V\|_1=\sum_{w\in W_q}|V(w)|$ and $\|V\|_{\infty}=\sup_{w\in W_q} |V(w)|.$ We  define $\Fc_{1,q}=\{V\in \C^{W_q}: \|V\|_1<\infty\}$ and $\Fc_{0,q}$ the set of functions $V\in \C^{W_q},$ with $ \#\{w\in W_q: V(w)\not=0\}<\infty.$  For any  $N\in\N^*,$  $U_1,\ldots, U_q\in \Un(N)$ and $V\in \Fc_{1,q}$  the following sum converges almost surely and defines a random matrix 
$$V(U_1,\ldots, U_q)= \sum_{w\in W_q} V(w) w(U_{f},1\le f\le q)\in M_N(\C).$$ 

\begin{theomss}{\textbf{\ref{Theom Energy Words}}} For $t\in\R_+^q $ and $V\in \Fc_{0,q}$, there exists 
$r_V>0$ and analytic functions $\vp_{t,V}, (\psi_{t,V,N})_{N\ge 1}$ and $\psi_{t,V}$ on $D_{r_V}=\{z\in\C: |z|<r_V\},$ such that 
$$ e^{\psi_{t,V,N}(z)}=\esp[e^{zN\Tr(V(U_{1,t_1},\ldots, U_{q,t_q}))- N^2\vp_{t,V}(z)}]\longrightarrow e^{\psi_{t,V}(z)},$$
as $N\to\infty,$ where the convergence is uniform on compact subset of $D_{r_V}$.\end{theomss}


\begin{proof}[Proof of Theorem \ref{Theom Energy Words}]For any function $V\in \Fc_{1,q}$,  let us define the matrix $V_{N,t}= V(U_{1,t_1},\ldots, U_{q,t_q})$ and  $I_{t,V,N}(z)=  N^{-2} \log \esp[e^{N z  \Tr(V_{t,N})}].$ The latter analytic function satisfies on a neighborhood of $0$, 
$$ I_{t,V,N}(z)=\sum_{w\in W_q} V(w)\vp_{t,N}(w) z + \sum_{\substack{m\ge 2\\ w_1,\ldots, w_m\in W_q }}\frac{\prod_{i=1}^mV(w_i)}{m!}\vp_{t,N}([(w_1,\ldots,w_m ),0_m]) z^{m}. $$
According to Proposition \ref{PropoSystDiff Cut and Join}, the summand of the two sums converge pointwise.  The summand of the first sum is bounded by $|z V(w)|,$  so that this sum converges absolutly towards $\sum_{w\in W_q} V(w)\vp_t(w) z.$ Each coefficient of the second power series is bounded by
 $$\sum_{\substack{w_1,\ldots, w_m\in W_q\\T\in \Cc_m}}\frac{\prod_{i=1}^m |V(w_i)|}{m!}T(w_1,\ldots,w_m)\le \max_{a\not=b\in \supp(V)}\{\la \overline{n}_a,\overline{n}_b\ra_t\}^{m-1} \frac{m^{m-2}}{m!} \|V\|_1^m.$$
It follows that $I_{t,V,N}(z)=\sum_{m\ge 1, w_1,\ldots, w_m\in W_q}\frac{z^m\prod_{i=1}^m V(w_i) }{m!}\vp_{t,N}((w_1,\ldots,w_m),0_m)$ is well defined  on $\Dd_{r_V}$ with $$\frac{1}{r_V}=\max_{a,b\in \supp(V)}\{\la \overline{n}_a,\overline{n}_b\ra_t\} e\|V\|_1$$ and converges uniformly as $N\to \infty$ on its compact subset towards a limit that we denote by $\vp_{t,V}(z)$.    Let us set for any $V\in \C^{W_q},$ 
$$\eta_V=\sup_{a\in \supp(V)}  \|\overline{n}_a\|_t^{2 } \sum_{w\in W_q} \lambda_t(w_i)|V(w_i)|. $$
For any $m\ge 1$ and $V$ with $\eta_V<\infty,$ according to Lemma \ref{Bound second order},  the sum 
 $$\psi_{N,m}(V)=\sum_{w_1,\ldots, w_m\in W_q}\frac{\prod_{i=1}^m V(w_i) }{m!}\psi_{t,N}((w_1,\ldots,w_m),0_m)$$ 
 is well defined and satifies
 $$|\psi_{N,m}(V)|\le    2^{m-1}\frac{m^{m}}{m!}\eta_V^{m}.$$
Thanks to Proposition \ref{PropoSystDiff Cut and Join}, $\psi_{m}(V)=\sum_{w_1,\ldots, w_m\in W_q}\frac{\prod_{i=1}^m 
V(w_i) }{m!}\psi_{t}((w_1,\ldots,w_m),0_m)$ is well defined and satisfies $\psi_{N,m}(V)\to \psi_{m}(V)$. What is more, if 
 $\psi_{t,V,N}(z)$ denotes the power series with coefficients   $(\psi_{N,m}(V))_{m\ge 1},$ then $\psi_{t,V,N}$ is well defined on 
 $D_{r'_V}$ and converges uniformly on its compact subset towards a function $\psi_{t,V},$ with $$\frac{1}{r'_V}= 2 e \eta_V.$$
To conclude, note  that   $r'_V>r_V,$ so that if $|z|<r'_V<\infty,$ $\vp_{t,V}(z)$ is well defined  and the analytic function 
$$e^{\psi_{t,V,N}(z)}= \esp[e^{zN\Tr(V_N)- N^2\vp_{t,V}(z)}]$$  
converges uniformly towards $e^{\psi_{t,V}(z)}$ on  $D_{r'_V}.$
\end{proof}
\begin{remark} It follows from (\ref{Formula Cumulants one Trace}) that if $x_1^{[m]}$ denotes the tuple  composed with  $m$ copies of $x_1$, and $(U_t)_{t\ge 0}$ a $\Un(N)$-Brownian motion, then
$$\vp_{t,N}([x_1^{[m]},0_m])=(m-1)!t_1^{m-1}\int_{\Delta^{m-1}} \esp[\Tr(U_{s_m t_1}U_{s_{m-1}t_1}\ldots U_{s_1t_1})]ds=  t_1^{m-1} e^{-\frac{t_1}{2}}.$$ Therefore,  if  $V=\delta_{x_1}$, for any $N\in \N^*$,  $$\vp_{t,N,V}(z)=\vp_{t,V}(z)=\frac{(e^{zt_1}-1)e^{-\frac{t_1}{2}}}{t_1}.$$
\end{remark}
For any function $V\in \C^{W_q}$ and any word $w=x_{i_1}^{\ep_1}\ldots x_{i_n}^{\ep_n}\in W_q$, with $\ep_1,\ldots, \ep_n\in \{-1,1\},$ let us set  $V^*(w)= \overline{V(x_{i_n}^{-\ep_n}\ldots x_{i_1}^{-\ep_1})}.$ We say that $V\in \C^{W_q}$ is \emph{symmetric} if $V^*=V$.  For any $N\in \N^*,$ let $(U_{1,t})_{t\ge 0},\ldots, (U_{q,t})_{t\ge 0}$ be $q$ independant $\Un(N)$-Brownian motions. For any symmetric function $V\in\Fc_{1,q}$ and $t\in \R_+^*,$ the random matrix 
$$V_{t,N}=V(U_{1,t_1},\ldots, U_{q,t_q}) $$  
is hermitian and its operator norm is bounded by $\|V\|_1.$ In particular, it satisfies $0<\esp[e^{N\Tr(V_{t,N})}]<\infty$. Let   $\mu_{t,V}$ be the probability measure on $\Un(N)^q$, whose density  with respect to the law of $(U_{1,t_1},\ldots,U_{q,t_q})$ is $\esp[e^{N\Tr(V_{t,N})}]^{-1}e^{N\Tr(V(U_1,\ldots,U_q))}.$  We shall  denote by $(U^V_{1,t_1},\ldots,U^V_{q,t_q})$  a random variable distributed as $\mu_{t,V}$ on  $\Un(N)^q.$ 

For any $V,W\in  \C^{W_q},$ let us define  for any $t\in \R_+^q,$
$$\Jc_{t,V}(W)= \sum_{m\ge 1, a,w_1,\ldots, w_m\in W_q,T\in \Cc_m} |W(a)|\frac{\prod_{i=1}^m|V(w_i)|}{m!}T_t(w,w_1,\ldots, w_m)$$
and 
\begin{align*}
\Jc_{t,2,V}(&W)=\sum_{a,b\in W_q} \la \overline{n}_a,\overline{n}_b\ra_t |W(a)||W(b)|\\& +\sum_{m\ge 1, a,b,w_1,\ldots, w_m\in W_q,T\in \Cc_m} |W(a)||W(b)|\frac{\prod_{i=1}^m|V(w_i)|}{m!}T_t(a,b,w_1,\ldots, w_m).
\end{align*}
For a fixed $V\in \C^{W_q}, t\in \R_+^q,$  we consider $\Fc_{t,V}=\{ W\in \Fc_{1,q}: \Jc_{t,1,V},\Jc_{t,2,V}<\infty    \}.$

\begin{theomss}{\textbf{\ref{MF Potential}}} Let   $V\in \Fc_{1,q}$.  Then, under the measure $\mu_{N,V}$, the family of random variables $\frac{1}{N}\Tr(W(U^V_{1,t_1},\ldots, U^V_{q,t_q}))$, with  $W\in \Fc_{t,V}$ converges jointly  in probability towards  constants $\Phi_{t,V}(W),$ $W\in \Fc_{t,V}.$
\end{theomss}


\begin{remark} For any Cayley tree $T\in \Cc_m,$ let us write $(d_T(i))_{1\le i\le m }$ for the degree distribution of $T$. For $m\ge 2,$ Cauchy-Schwarz inequality yields that for $w_1,\ldots, w_m\in W_q,$
\begin{align*}
\frac{\prod_{i=1}^m|V(w_i)|}{m!}T_t(w_1,\ldots, w_m) \le\sum_{\substack{T\in \Cc_m\\ w_1,\ldots, w_m\in W_q}} \frac{1}{m!}  \prod_{i=1}^m|V(w_i)|\|\overline{n}_{w_i}\|^{d_T(i)}.
\end{align*}
For any $V\in \C^{W_q},$  according to formula (1) of \cite{BM},  
$$\sum_{w_1,\ldots,w_m\in W_q}\prod_{i=1}^m|V(w_i)|T_t(w_1,\ldots, w_m) \le \sum_{\substack{ w_i\in W_q}} \prod_{i=1}^m\|\overline{n}_{w_i}\|_t|V(w_i)|  (\sum_{i=1}^m\|\overline{n}_{w_i}\|)^{m-2}.$$
In particular,  $\Jc_{1,V}(W),\Jc_{2,V}(W)<\infty,$  if $V,W\in \Fc_{0,q},$ with $$\|V\|_{\infty} .\sup_{\substack{a\in W_q\\ W(a) \text{ or }V(a)\not=0}} \|\overline{n}_a\|_t< \frac{1}{e}.$$   \end{remark}

\begin{proof} Let $W,V\in \Fc_{1,q}$, with $\Jc(V)<\infty,$ the  sum of random matrices $W_{t,V}= \sum_{a\in W_q}W(w) w(U^V_{1,t_1},\ldots, U^V_{q,t_q}),$ converges almost surely.    Then, $\esp[\frac{1}{N}\Tr(W_{t,V})]$  and $\Var(\Tr(W_{t,V}))$ are respectively equal to the following  absolutly convergent sums, 
$$ \sum_{w\in W_q}W(w)\vp_{t,N}(w) +   \sum_{\substack{m\ge 1,w,w_1, \\ \ldots, w_m\in W_q}}\frac{1}{m!}W(w) \prod_{k=1}^mV(w_k) \vp_{t,N}([(w,w_1,\ldots, w_m), 0_{m+1}])$$
and
\begin{align*}
 \sum_{a,b\in W_q}W(a)&W(b)\vp_{t,N}([(a,b),0_2]) \\
 &+\sum_{\substack{m\ge 1,w,w_1, \\ \ldots, w_m\in W_q}}\frac{1}{m!}W(a)W(b) \prod_{k=1}^mV(w_k) \vp_{t,N}([(a,b,w_1,\ldots, w_m), 0_{m+2}]).
\end{align*}
According to Proposition \ref{Cumulant's control}, dominated convergence implies  that these two sequences have a limit as $N\to\infty.$
\end{proof}


\vspace{0,5 cm}


\subsection{Central limit theorem}  As a consequence of Proposition \ref{PropoSystDiff Cut and Join}, we get the
 
\begin{prop} For any $t\in\R_+^q,$ the family $(\Tr(w_{t,N})- N\vp_{t}(w))_{w\in W_q}$ converges towards the centered gaussian field $(\phi_w)_{w\in W_q}, $ such that  for any $a,b\in W_q,$ $\cov(\phi_w,\phi_{w'})= \vp_{t}([(w,w'),0_2]).$
\end{prop}
\begin{proof} For any word $w\in W_q,$ $\esp[N^{-1}\Tr(w_{t,N})]= \vp_{t}(w)+O(N^{-2}),$  whereas for any $m\ge 2,$ $C_m(\Tr(w_{1,t,N}),\ldots,\Tr(w_{m,t,N}))= N^{2-m}\vp_{t,N}([(w_1,\ldots, w_m),0_m]) $ converges,  as $N\to\infty,$ towards $\vp_{t}([(w_1,w_2),0_2]),$  if $m=2$, and $0,$ if $m\ge 3.$ 
\end{proof}
Let us remark that  in the proof of Proposition \ref{Cumulant's control},  we obtained in formula (\ref{Formula Cumulants one Trace}) an expression of the function $\vp_{t,N}$ for any $N\in\N^*,$   in terms of its restriction to single words. Specialized to partial words with two blocks, this gives an expression of  the covariance of the above field.  Let  us define a family of $3q$  independant  $\Un(N)$-Brownian motions $(U_{1,t})_{t\ge 0}, (V^{1}_{1,t})_{t\ge 0},(V^{2}_{1,t})_{t\ge 0},\ldots, (U_{q,t})_{t\ge 0}, (V^{1}_{q,t})_{t\ge 0},(V^{2}_{q,t})_{t\ge 0} $. For any $w_1,w_2\in W_q$ and $(a,b)\in\Nc_{2, (w_1,w_2)}(f)\setminus \Sc_{2,(w_1,w_2),0_2}, $  let $\chi:\{1,\ldots, \ell(w_1w_2)\}\to\{1,2\}$ be the function such that the $i^{\text{th}}$ letter of $\Tc_{a,b}((w_1,w_2))$ belongs to $w_{\chi(i)}.$ If $\Tc_{a,b}((w_1,w_2))= x_{i_1}^{\ep_1}\ldots x_{i_n}^{\ep_n}$, let us set for any $r,s\in \R_+^q,$ 
 $$w_{a,b}(r,s)=\left(U_{i_1}V^{\chi(1)}_{i_1}\right)^{\ep_1}\ldots \left(U_{i_n}V^{\chi(n)}_{i_1}\right)^{\ep_n}.$$
According to formula (\ref{Formula Cumulants one Trace}),
 $$ \cov(\Tr(w_{1,t,N}),\Tr(w_{2,t,N}))=\sum_{1\le f\le q ,\ep\in \{-1,1\}}\ep t_f \int_{0}^1 \sum\frac{1}{N}\esp[\Tr(w_{a,b}(st,(1-s)t))]ds,$$
where the second sum is over pairs $(a,b)\in\Nc^\ep_{2, (w_1,w_2)}(f)\setminus \Sc_{2,(w,w'),0_2}$.

%Let us denote by $\Kc_t (x,y)$ and $L_t(x,y)$ the power series 
%$$\Kc_t(x,y)=G_t^+(e^{-\frac{t}{2}}x,e^{-\frac{t}{2}}y)+G_t^+(\frac{e^{-\frac{t}{2}}}{x},\frac{e^{-\frac{t}{2}}}{y}) +G^-_t(e^{-\frac{t}{2}}x,\frac{e^{-\frac{t}{2}}}{y})+G^-_t(\frac{e^{-\frac{t}{2}}}{x},e^{-\frac{t}{2}}y)$$


%\noindent and $$ L_t(x,y)=xy�_x�_y \Kc_t(x,y).$$
%The power series $L_t(x,y)$ is the generating function of $(k_t(n,m))_{n,m\in\Z}$.   Observe that it follows easily from  (\ref{genplus}) and (\ref{genmoins}) that $$\Kc_t(x,y)\to -\log\left((1-\frac{x}{y})(1-\frac{y}{x})\right),$$
%as $t\to \infty.$ 




\section{Planar Yang-Mills measure\label{Yang-Mills}}
We shall see in this section that representing words as loops in the plane allows to define and study the previous models in one same framework.  In the next section, we shall then give analog of Schwinger-Dyson's  equations.

\subsection{Paths of finite length}  Let us first specify the family of loops we are considering. Let us call  \emph{parametrized path} any   Lipschitz function from $[0,1]$ to $\R^2$, that are either constant or with speed bounded by below.   We denote  by  $\Pd(\R^2)$  the set of parametrized paths up to  bi-Lipshitz increasing reparametrization and call its elements \emph{paths}.   For any path $\gamma\in \Pd(\R^2)$ with parametrization $c:[0,1]\to\R^2$, let us denote its endpoints $c(0)$ and $c(1)$ by $\underline{\gamma}$ and $\overline{\gamma}$, and by $\gamma^{-1}$ the reverse path parametrized by $t\in[0,1]\mapsto c(1-t)$. 
For any $x\in\R^2$, we  denote  by $\Ld_x(\R^2)$ the set of paths $\gamma\in\Pd(\R^2)$ such that $\underline{\gamma}=x=\overline{\gamma}$ and call elements of $\Ld_x(\R^2)$ loops based at $x$.  We set $\Ld(\R^2)= \cup_{x\in \R^2}L_x(\R^2)$. For any loop $l$ based at some point $x\in\R^2$ and parametrized by the Lipschitz-continuous map $\tilde{l}: [0,1]\to \R^2$, we call \emph{non-based loop} the induced map $\U\to\R^2$ up to  bi-Lischitz order preserving one-to-one mapping  of $\U$. If $a$ and $b$ are two paths such that $\overline{a}=\underline{b}$, we denote by $ab$ the path of $\Pd(\R^2)$ obtained by concatenation.  


\subsection{Embedded graphs\label{section embedded graphs}} We call here \emph{ embedded graph in the plane}  the data of  a triple  of finite sets $\mathbb{G}=(\Vbb,\Ebb,\Fbb_b)$, where $\Fbb_b$ are simply connected domains of the plane with disjoint interior, which boundary is the image of a non-based loops, $\Ebb$ is a set of paths of $\Pd(\R^2)$ such that the union of their image  is the union of boundaries of elements of $\Fbb_b$, $\Vbb$ is the set of  endpoints of $\Ebb$ and  the graph induced by $\Ebb$ on $\Vbb$ is connected. With this convention, any edge $e\in \Ebb$ is either a simple loop or an injective path  of finite length. We denote by $F_{\infty,\Gbb}$ (or simply $F_\infty$)  the interior of $\R^2\setminus (\cup_{F\in \Fbb_b} \bar{F})$. We set  $\Fbb= \Fbb_b\cup \{F_\infty\}$ and     denote by $|F|$ the area of any element of   $F\in\Fbb_b$ and by $�F$ the non-based loop whose  image is the boundary of $F$ oriented counterclockwisely.     We shall write $\Pd(\Gbb)$ for  the set of  paths  that are concatenation of elements of $\Ebb$ and $\Ld(\Gbb)$ (and resp. for any $v\in \Vbb$, $\Ld_v(\Gbb)$) for the set of loops (resp. loops based at $v$)  in $\Pd(\Gbb).$ 


\subsection{A free group: reduced loops of an embedded graph}

Let us fix an embedded graph $\Gbb$. For any  pair of  paths $\gamma_1$ and $\gamma_2$ of $\Pd(\Gbb)$,  let us write
 $\gamma_1\sim \gamma_2$ and say  that  $\gamma_1$ and $\gamma_2$  are equivalent  if one can get $\gamma_1$ from 
 $\gamma_2$ or vice-versa by adding or erasing  paths of the form $e.e^{-1}$, with $e\in\Ebb.$ For any path $\gamma$, there is a unique 
 element of minimal length   in its equivalence class,  that the call the \emph{reduction} of $\gamma$. The set of reduced paths endowed 
 with the operation of concatenation and reduction forms a groupoid that we denote by $\RP(\Gbb).$  For any $v\in \Vbb$, we denote by 
 $\RL_v(\Gbb)$ the set of reduced paths that are loops based at $v$. Endowed with the above multiplication, $\RL_v(\Gbb)$ is a free group 
 of rank $\#\Fbb_b$   (we shall highlight specific free basis in section \ref{Basis RL}).   
 
 
 \subsubsection{Multiplicative functions}

For any $N\in\N^*,$ we call a   function $h:\Ld_v(\R^2)\to \Un(N),$  \emph{multiplicative} if for any paths  $a,b\in \Ld(\R^2)$, with $\underline{a}=\underline{b},$  $$h(a b  )=h(a)h(b). $$
We denote the space of multiplicative functions by $\Mca(\Ld(\R^2), \Un(N))$ and by $\Cc$ the smallest $\sigma$-fields such that for any $l\in \Ld(\R^2)$, $h\in \Mca(\Ld(\R^2),\Un(N))\longmapsto h(l) \in U(N)$ is measurable, where $U(N)$ is endowed with its Borel $\sigma$-fields. We define   as well, for any $v\in\Vbb $, the set  $\Mca(\Ld_v(\Gbb),U(N))$ of multiplicative functions on $\Ld_v(\Gbb)$.  For any choice of of basis $\Lambda$ of $\RL_v(\Gbb)$, there is a bijection  
\begin{align*}
\Theta_\Lambda:\Mca(\Ld_v(\Gbb),U(N))\to\Un(N)^{\#\Fbb_b}\\
h\mapsto (h(\lambda))_{\lambda\in\Lambda}.
\end{align*}
 It is easy to see that the preimage $\sigma$-field of the Borel $\sigma$-field by this map does not depend on this choice and we denote it by $\Cc_\Gbb.$
\subsubsection{Lassos basis and discrete Yang-Mills measure\label{Section Discrete YM}}
For any loop $v\in \Vbb$ and $l\in \Ld_v(\Gbb)$, we say that $l$ is a lasso based at $v$, if  $l= a\partial_{\overline{a}}F a^{-1}$ where $a\in\Pd(\R^2)$, with $\underline{a}=v$, $\overline{a}$ is a vertex in the image of $\partial F$  and     $\partial_{\overline{a}}F $ is the rooting of $\partial F$ at $\overline{a}.$ We shall see in the next section that there exists basis of $\RL_v(\Gbb)$ formed with lassos.  Let  $\Lambda=(\lambda_F)_{F\in \Fbb_b}$ be a lassos free basis of $\RL_v(\Gbb)$, and  let  $YM^\Lambda_\Gbb$  be the law of $\Theta_\Lambda^{-1}( (U_{F,|F|})_{F\in\Fbb_b})$   on $(\Mca(\Ld_v(\Gbb),\Un(N)),\Cc_\Gbb),$ where $((U_{F,t})_{t\ge 0})_{F\in \Fbb_b}$ is a family of independant Brownian motions on $\Un(N).$
\begin{lemma}[\cite{MF,Franck}] \label{Discrete YM}i) For any lassos free basis $\Lambda,\Lambda'$ of $\RL_v(\Gbb),$ $YM^\Lambda_\Gbb=YM^{\Lambda'}_{\Gbb}.$ We denote this law by $YM^v.$

ii) If $\Gbb'$ is another embedded graph with $\Pd(\Gbb')\subset \Pd(\Gbb)$ and $\Rc^{\Gbb}_{\Gbb'}:\Mca(\Ld_v(\Gbb))\to \Mca(\Ld_v(\Gbb'))$ denotes the restriction map, then $${\Rc^{\Gbb}_{\Gbb'}}_*YM^v_\Gbb=YM^v_{\Gbb'}.$$ 
 \end{lemma} 
We denote by $(H_l)_{l\in \Ld_v(\Gbb)}$ the canonical process on $\Mca(\Ld_v(\Gbb),\Un(N))$ with law $\YM_\Gbb.$ The first point follows from the invariance of the law of the $\Un(N)$-Brownian motion by adjunction and the following result.
\begin{theomss}[\cite{HUMPHRIES}] Let  $X=(x_1,\ldots, x_n)$ and $Y=(y_1,\ldots ,y_n)$ be two free basis for the free group $F_n$, such that $x_i$ is conjugated to $y_i$ for all $i\in \{1,\ldots,n\}$.  Then  $X$ can be obtained from  $Y$ by a sequence of transformations of the kind   $(u_1,\ldots , u_n)\mapsto (u'_1,\ldots, u'_n)$ where, for some $i,j$,   $u'_i=u_ju_i u_j^{-1}$ or $u_j^{-1}u_i u_j$
and $u'_k=u_k$ for $k\not=i$. 
\end{theomss}
The second point of Lemma \ref{Discrete YM} requires a proof (see \cite{MF, Franck}) that we won't reproduce here; an argument goes as follows. Let  $\Gbb'=(\Vbb',\Ebb',\Fbb_b' )$ be an embedded graph with $v\in \Vbb'$ and $\Pd(\Gbb')\subset\Pd(\Gbb)$. Assume that there exists $F\in \Fbb_b'$ and $F_1,F_2\in \Fbb_b$ with $\overline{F}=\overline{F_1\cup F_2}.$    If  $\lambda$ and  $\lambda'_{1},\lambda'_{2}$  are lassos in $\Ld_v(\Gbb')$ and $\Ld_v(\Gbb)$, with faces $F,F_1,F_2$ such that $\lambda= \lambda_1\lambda_2$, then under $YM_\Gbb$, $H_{\lambda}$  has the same law as $U_{1,|F_1|}U_{2,|F_2|},$ where $(U_{1,t})_{t\ge 0}$ and $(U_{2,t})_{t\ge 0}$ are two independant $\Un(N)$ brownian motions. Hence, it has the same law as $U_{1,|F|}$, that is the law of $H_\lambda$ under $YM_{\Gbb'}.$
\subsection{Yang-Mills measure}
 Let $d_1$ and $d_l$  be the two distances on $\Pd(\R^2)$ defined in the following way: for any pair of paths $\gamma_1,\gamma_2\in \Pd(\R^2)$, parametrized by $c_1,c_2:[0,1]\to \R_2,$

$$d_1(\gamma_1,\gamma_2)=|\underline{\gamma_1}-\underline{\gamma_2}|+ \int_0^1 |c'_1(t)-c'_2(t)|dt$$
and
$$d_\ell(\gamma_1,\gamma_2)=\inf_{\phi,\psi }\sup_{r,s\in [0,1]}\{ |c_1\circ\phi(r)-c_2\circ\psi(s)|\}+|\ell(c_1)-\ell(c_2)|,$$
where we have denoted by $\ell(c)$ the length of a path $\gamma\in\Pd(\R^2)$ and the infimum is taken over all increasing bijections of $[0,1]$. It have been proved in  \cite{Champsmarkoholo} that $d_1$ and $d_\ell$ induce the same topology on $\Pd(\R^2)$, though $(\Pd(\R^2),d_1)$ is complete and $(\Pd(\R^2),d_\ell)$ is not. In the following, we shall only use this topology on closed space $\Ld_0(\R^2)$ and say that a sequence of paths $(l_n)_{n\ge 0}$ converges to $ l$ if   $d_\ell(l_n,l)\to 0$ and $\underline{l_n}=\underline{l}$ for every $n\in\N^*$.  For any embedded graph $\Gbb,$ with $v$ as a vertex,  let us denote by $\Rc^v_\Gbb:\Mca(\Ld(\R^2),\Un(N))\to \Mca(\Ld_v(\Gbb),\Un(N))$  the restriction mapping. This application is measurable with respect to the $\sigma$-fields $\Cc$ and $\Cc_{\Gbb}$. It is shown in \cite{Champsmarkoholo} that the measures $YM_\Gbb$, with $\Gbb$ embedded graphs, can be extended in the following way.
\begin{theom} \label{continuiteYM}there exists a probability measure $YM_N$ on $(\Mca(\Ld(\R^2), \Un(N)), \Cc )$ such that, for any embedded graph $\Gbb$, $v\in\R^2,$ $${\Rc^v_{\Gbb}}_{*}YM_N  =YM_\Gbb.$$
Let $(H_l)_{l\in \Ld(\R^2)}$ be a random multiplicative function   with law  $YM_N$. If $(l_n)_{n\ge 0}$ is a sequence of paths in $\Ld(\R^2)$ that converges to $l$, then, under  $YM$, $H_{l_n}$ converges in probability towards $H_l$. If $h$ is an area preserving diffeomorphism of $\R^2,$ then the process $(H_{h(l)})_{l\in \Ld(\R^2)}$ and $(H_l)_{l\in \Ld(\R^2)}$ have the same law.
\end{theom}
For any $l\in \Ld(\R^2)$ and $N\in \N^*$, the random variable $\frac{1}{N}\Tr(H_l)$ is called a \emph{Wilson loop}. 

\subsection{\texorpdfstring{$U(1)$}{Lg}-Yang-Mills measure}

Let us consider the commutative case, $N=1$. Let  $\Gbb$ be an  embedded graph in the plane, $v$ a vertex of $\Gbb$. For any loop $l\in \RL_v(\Gbb)$,  its winding number function define an element $n_l\in \Ld^2(\R^2)$. Let us fix a family of lassos $(\lambda_F)_{F\in \Fbb_b}$ of $\Gbb$. Under $\Un(1)$-YM measure,  $(H_{\lambda_F})_{F\in\Fbb_b} $ has the same law as  $\#\Fbb_b$ independent marginals of $\Un(1)$-Brownian motion $(U_{F, |F|})_{F\in \Fbb_b}$. Let $W$ be a white noise on the  plane with intensity given by the  Lebesgue measure. The random family   $(H_l)_{l\in \RL_v(\Gbb)}$ is equal to $(\prod_{F\in \Fbb_b}H_{\lambda_F}^{n_F(l)})_{l\in \RL_v(\Gbb)}$ and has the same law as $(\exp\left(iW(n_l)\right))_{l\in \RL_v(\Gbb)}$.  For any loop $l\in \Ld(\R^2)$, according to Banchoff-Pohl inequality (see Lemma \ref{Banchoff-Pohl}), its winding number defines an element  $n_l\in \Ld(\R^2).$ Moreover, according to Theorem 3.3.1. of \cite{Champsmarkoholo}, the map $l\in \Ld_0(\R^2)\to \Ld^2(\R^2)$ is continuous, so that, if $(l_n)_{n\ge 0}$ is a sequence of $\Ld_0(\R^2)$  that converge for $d_1$ topology to a loop  $l\in \Ld_0(\R^2)$,  the sequence of  random variables $\exp\left(i W(n_{l_n})\right)$ converges to $\exp\left(i W(n_{l})\right)$ in distribution. Hence, the process $(H_l)_{l\in \Ld(\R^2)}$ introduced in Theorem \ref{continuiteYM} has the same law as $(\exp\left(i W(n_{l})\right))_{l\in\Ld(\R^2)}.$ Moreover,  the same argument and Lemma \ref{det} yield the following Lemma.
\begin{lemma} For any integer $N\in\N^*$, under $YM_N$,  the law of $(\det(H_l))_{l\in \Ld_0(\R^2)}$  and $(\exp\left(i W(n_{l})\right))_{l\in \Ld_0(\R^2)}$ is $YM_1$. 
\end{lemma}
\subsection{Two free basis of the  group of reduced loops\label{Basis RL}}
 We shall present two families of free basis of $\RL_{v}(\Gbb)$.  Let $\Ebb^+$  be an orientation of $\Gbb$, that is a subset of $\Ebb$ such that for any $e\in\Ebb$, $e$ or $e^{-1}\in\Ebb^+$. Let us also fix  a spanning tree $T$ of the graph $\Gbb$ and set $T^+$ the collection of positively oriented edges of $T.$ We denote by $e:\Fbb_b\to \Ebb^+\setminus T^+$ the unique bijection such that for any face $F\in \Fbb_b$, $e(F)$ is bounding the face  $F$.  For any $e\in \Ebb$, bounding a face $F$,  we denote by $�_eF$ the loop starting with $e$ and bounding $F.$ For any  $x,y\in \Vbb$, we denote by $[x,y]_T$ the unique path in $T$ going from $x$ to $y$.   Let us now define two families of loops by setting for any  edge $e\in\Ebb,$

$$\beta_e=  [v,\underline{e}]_T e [\overline{e},v]_T$$
and for any face $F\in \Fbb_b$,
 $$ \lambda_F= [v,\underline{e(F)}]_T�_{e(F)}F [\underline{e(F)},v]_T.$$ 
 
 \noindent It is easy to see that $\RL_{v}(\Gbb)$ is a free group of rank $\#\Fbb_b$    with free basis $(\beta_e)_{e\in \Ebb^+\setminus T^+}$.  For any loop $l\in \Ld(\Gbb)$, 
 \begin{equation}\label{decompositionenbeta}
l \sim\beta_{e_1}\beta_{e_2}\cdots \beta_{e_n},
\end{equation}
 where $e_1, \ldots, e_n$ are the edges in $\Ebb\setminus T$,  used by the loop $l$ in this order. In \cite{MF}, it is proved that the second family of loops  is another free basis of $\RL_{v}(\Gbb)$.
  \begin{lemma}[\cite{MF}] The family  $(\lambda)_{F\in\Fbb_b}$ is a free basis of $\RL_{v_0}(\Gbb)$  \end{lemma} 
%such that  the pushforward of $YM^\Gbb$ is $$\ts_{e\in T^+}dg_e\ts_{F\in \Fbb_b}q_{t(F)}(g_F)dg_F .$$ 
Let us give  the change of basis between the two basis of $\RL_{v}( \Gbb)$ defined above.  Denote by $\hat{\Gbb}=(\hat{\Vbb},\hat{\Ebb})$ the  graph dual to  $\Gbb$:  its vertices   are indexed by the set of faces $\Fbb$, whereas its edges are indexed by the set $\Ebb$ such that two faces $F_1$ and $F_2$ of $\Gbb$ are neighbours  in $\hat{\Gbb},$ if their boundaries share a common edge.   For any edge $e\in \Ebb$,  we denote by  $F_L(e)$ and $F_R(e)$ the edges on the left and on the right  of $e$ and denote by $\hat{e}$ the edge  $(F_L(e),F_R(e))\in \hat{\Ebb}$.   Let $\hat{T}=\Ebb\setminus T$ be the dual spanning tree of $T$, considered as rooted at the infinite face $F_\infty.$   We fix  an  orientation $\Ebb^+$ of $\Gbb$, such that for any edge  $e\in\Ebb^+\setminus T$, the distance in  $\hat{T}$ to the root $F_\infty$  decreases along $\hat{e}$. Note that for any bounded face $F$,   $F_L(e(F))=F$. For any  face $F$, we denote by $\hat{T}_F$ the subtree of $\hat{T}$ with root $F$ and with vertices  the set of childs of $F$ in $\hat{T}$, that we denote by $C_F$.  For any edge $e\in \Ebb^+\setminus T^+,$ denote by   $\lec_e$   the order  on $\hat{T}_{F_L(e)}$ induced by the  time of the first visit by the \emph{clockwise contour process} boarding the dual tree $\hat{T}$ starting along the left of $\hat{e}^{-1}$, as is displayed with an example in figure \ref{DecompositionLassos}.  Then, for any edge $e\in \Ebb^+\setminus T$,
$$\lambda_{F_L(e)}=\beta_{e}\left( \overset{\longrightarrow}{\prod}_{F\in C_{F_L(e)}} \beta_{e(F)}\right)^{-1}$$
and

\begin{equation}
  \beta_e= \overset{\longrightarrow}{\prod}_{ F\in  \hat{T}_{F_L(e)} } \lambda_F, \label{lassobeta}
\end{equation}
where $\overset{\longrightarrow}{\prod}$ denotes the product of terms increasing for $\lec_e$, from the left to the right.  For any loop $l\in \Ld_v(\Gbb)$, we  denote by $w_l$ the word with letters  $(\lambda_F)_{F\in \Fbb_b}$  and their inverse such that $l\sim w$, given by the decomposition  (\ref{decompositionenbeta})   and the inversion formula (\ref{lassobeta}). Using notation of section \ref{Section Partial words}, for any face $F\in\Fbb_b$ and any complex number $z\in F$, $n_{w_l}(F)=n_l(z).$



\begin{figure}\centering \includegraphics[height=3in]{DecompositionLassos}
\caption{\label{DecompositionLassos}We represent with black lines a spanning tree of a square grid together with its dual with dashed red lines. We also display an   edge $e$ of $\Ebb^+\setminus T$ in blue together with the order $\lec_e$ on $C_{F_L(e)}$,  by numbering its elements and drawing in black the clockwise contour process around $\hat{T}_{F_L(e)}$.}\end{figure}

\subsubsection{Complexity of lassos decompositions:} We can now give an estimate on the complexity of  the decomposition of a loop in $\Gbb$  in a word of lassos associated to a spanning tree $T$.  We display in that section results of \cite{MF} in a slightly different form adapted to our purpose. For any subset $E\subset \Ebb$ and any loop $l\in \Ld(\Gbb)$, denote by $\Lc_E(l)$ the number of times that $l$ uses the edges of  $E$ or $E^{-1}$.


\begin{lemma} \label{indicemaxcombi}Let $l\in \Ld(\Gbb)$  be a loop of $\Gbb$. Then, for any face $F\in \Fbb_b$, $$\bar{n}_{\lambda_F}(w_l)= \Lc_{[F,F_\infty]_{\hat{T}}}(l).$$
\end{lemma}
\begin{dem} The decomposition  (\ref{decompositionenbeta}) and (\ref{lassobeta}) yield the equality.
\end{dem}


 From  now on, we shall choose the spanning tree in the following way.

 \begin{lemma} there exists a spanning tree $T$ of $\Gbb$ such that for any face $F\in\Fbb_b$, $$d_{\hat{T}}(F,F_\infty)=d(F,F_\infty).$$
 \end{lemma}
 For any loop $l$ of an embedded graph, we shall  control the maximal amperean area $A_t(w_l)=\sum_{F\in\Fbb} |F|\bar{n}_{w_l}(F)^2$ with the length of the loop $\ell(l)$.   For any loop $l\in \Ld(\R^2)$, denote by $n_l$ the winding number function of $l$. The \emph{Amperean area } of $l$ is the integral   $$A(l)=\int_{\R^2}n_l(x)^2dx.$$
 \begin{lemma}[Banchoff-Pohl inegality,\cite{BP}] \label{Banchoff-Pohl}For any loop of finite length $l\in \Ld(\R^2)$, $$A(l)\le \pi \ell(l)^2.$$
 \end{lemma}
 Note that if   $\bar{n}_l=\pm n_l\in  \Z^\Fbb$, that is, if $l$ winds only  to the left or only to the right, then the Banchoff-Pohl inegality  gives  the  expected bound. To treat more general loops, we need the following lemma.
 \begin{lemma}  \label{lacetmax}there exists a loop $\bar{l}\in \Ld(\Gbb)$, which  does not use any edge twice, such that for any face $F\in \Fbb$ and $z\in F,$  $$ n_{\bar{l}}(z)=d(F,F_\infty).$$    
 \end{lemma}
 
 \begin{lemma}\label{borneAmpcombi}  Fix $p\in \N^*$ and loops $l_1,\ldots ,l_m \in\Ld(\Gbb)$  in an embedded graph $\Gbb$ such that  the union of the edges of $l_1, \ldots , l_m$ is $\Ebb.$ Let $l\in\Ld(\Gbb)$ be a loop that uses each edge at most $p$ times. Then $$A(w_l)\le   � p^2(\ell(l_1)+ \ldots +\ell(l_m))^2. $$
 \end{lemma}
 \begin{dem} The   assumptions together with  Lemma \ref{indicemaxcombi} yield that for any face $F\in\Fbb_b$, $$\bar{n}_{w_l}(F)\le p .d(F,F_\infty). $$
 Let us now choose a loop $\bar{l}$ as in Lemma \ref{lacetmax}. Then, $A_t(w_l)\le p^2 A(\bar{l}) $,    $\ell(\bar{l})\le \ell(l_1)+\ldots+\ell(l_m)$ and Banchoff-Pohl inequality yields  the expected bound. \end{dem}
 
\subsection{Asymptotics of Wilson loops as \texorpdfstring{ $N\to\infty$}{Lg}}



We call a finite  family  of loops  of $\Ld(\R^2)$ a \emph{skein}.  We denote  the set of skein by $\Sk(\R^2)$ and we endow it with the topology associated to the product topology.  If   $\Sc=\{l_1,\ldots, l_m\}$ is a skein, let us set $\Sc^*=\{l_1^{-1},\ldots, l_m^{-1}\}$ and for any $N\in \N^*,$ 
$$\Phi_N(\Sc)=N^{m-2}C_m(\Tr(H_{l_1}),\ldots, \Tr(H_{l_m})),$$
where the cumulants are with respect to the measure $YM_N$.  Observe that the law of the unitary Brownian motion is invariant under complex conjugation. Hence, for any skein $\Sc\in \Sk(\R^2)$ $$\Phi(\Sc)=\overline{\Phi(\Sc)}=\Phi_m(\Sc^*) $$  is real-valued. 
Let us denote by $\Ec_\Ac$  the set of  skeins of piecewise affine loops of $\R^2$ with transverse intersections of multiplicity at most $2$. We call elements of $\Ec_\Ac$ \emph{affine skeins}.  
\begin{prop} For any affine skein  $\Sc\in \Ec_{\Ac}$,  the sequence $\Phi_N(\Sc)$ converges  as $N\to \infty$. We denote its  limit by $\Phi(\Sc).$ 
\end{prop}
\begin{proof} For any affine skein $\Sc$, there exists an embedded graph $\Gbb$ such that the element of $\Sc$ belongs to $\Ld(\Gbb). $ Choosing an arbitrary base point $v\in \Vbb$ and decomposing each loops in a lassos basis yields that under $YM_N$, the random family $(H_l)_{l\in\Sc}$ has the same law a collection of words in marginals of independant $\Un(N)$ Brownian motions. Therefore, the Proposition  \ref{PropoSystDiff Cut and Join} implies the result.
\end{proof}


\begin{prop}  \label{UNIFOr}Let us fix a constant $K>0$. For any skein $\Sc\in\Ec_\Ac$ of loops of length smaller than $K>0$ and taking their values in a ball center  of radius $K$, $$ |\Phi_N(\Sc)-\Phi(\Sc)|\le \frac{36 m^2K^2}{N^2}e^{36� m^3K^2}.$$
\end{prop}

\begin{dem}   Let us assume that $\Sc=\{l_1,\ldots, l_m \}$ is a family of loops in $\Ec_\Ac$ all based at $0$.  Let  $\Gbb_\Sc=(\Vbb_\Sc,\Ebb_\Sc,\Fbb_\Sc)$ be the embedded graph with vertices the set of intersection points of the elements of $\Sc$ and with edges the restriction of elements of $\Sc$ between points of intersection.  Lemma \ref{borneAmpcombi} and the second inequality of Lemma \ref{Exponential Bound Amperean area}, for $k=2,$ imply that 

\begin{equation} \label{vitconvbase}
|\Phi_N(\Sc)-\Phi(\Sc)|\le \frac{4�}{N^2}(\ell(l_1)+ \cdots +\ell(l_m))^2e^{4� m (\ell(l_1)+ \cdots +\ell(l_m))^2}. 
\end{equation}
Consider now $(l_1,\ldots,l_m)\in \Ec_\Ac$ satisfying the assumption of the Theorem. For any $\ep>0,$ let us choose piecewise affine paths $\gamma_1,\ldots,\gamma_m$ such that $\underline{\gamma^i}=\underline{l_i}, \overline{\gamma_i}=0, \ell(\gamma_i)\le K(1+\ep)$ for any $i\{1,\ldots,m\}$ and $\{\gamma_i l_i\gamma_i^{-1},i\in \{1,\ldots,m\}\}$ is an affine skein. The application of (\ref{vitconvbase}) to $\{\gamma_i l_i\gamma_i^{-1},i\in \{1,\ldots,m\}\}$ yields the announced inequality.
\end{dem}
%\marginpar{\textcolor{red}{Commentaire meilleures bornes}}

% (\ell(l_1)+\ell(l_2)+\ldots+ \ell(l_m))^2e^{m(\ell(l_1)+\ell(l_2)+\ldots+ \ell(l_m))^2}
%Let $(l_1^n)_{n\ge 0},\ldots, (l_m^n)_{n\ge 0}$ be $m$ sequences of loops in $\Ld(\R^2)$ with a length uniformly bounded that converge to $l_1,\ldots, l_m\in\Ld(\R^2)$.
  %\marginpar{\textcolor{red}{Espace de Fock bosonique}}
 This result allows then to extend the function $\Phi$ to all of $\Sk(\R^2).$ Surprisingly, an argument analog to the proof Theorem 5.14.  of \cite{MF} applies as well to the higher order case.
 \begin{theom}\label{ConvCMSUP} For any skein $\Sc\in\Sk(\R^2)$, the sequence $\Phi_N(\Sc)$ converges as  $N\to \infty$. We denote its limit by $\Phi(\Sc)$. The function $\Phi$ is a real-valued   continuous function  on $\Sk(\R^2)$. If $h$ is an area-preserving diffeomorphism of $\R^2$, for any $\Sc\in \Sk(\R^2),$ $\Phi(h(\Sc))= \Phi(\Sc).$
 \end{theom}
 We shall also call the function $\Phi: \Sk(\R^2)\longrightarrow \R $  \emph{planar master field}. 
 
 \begin{proof}For any $K>0,$ let $\Sk_K$ (resp. $\Ec_K$) be the set of affine skeins with elements included in the ball of radius $0$ and with length less than $K$.  As $\cup_{K>0}\Sk_K=\Sk(\R^2),$ it is enough to prove the result on $\Sk_K.$ The set $\Ec_K$ is dense in $\Sk_K.$  Indeed, any loops $\Ld(\R^2)$ can be approximated by its linear interpolation, which itself can be approached by affine loop with a finite number of intersections. According to Theorem  \ref{continuiteYM}, for any $N\ge 1,$ the function $\Phi_N$ is  continuous on  $\Sk_K.$ Moreover, Proposition \ref{UNIFOr} shows that $\Phi_N$ converges uniformly towards $\Phi$ on $\Ec_K.$ Therefore, $\Phi_N$ converges uniformly towards the unique continuous extension $\tilde{\Phi}$ to $\Sk_K$ of $\Phi.$    \end{proof}
 
 A consequence of this Theorem is that for any $m\ge3,$ and any loops $l_1,\ldots , l_m$ in $\Ld(\R^2)$,   under $YM_N,$   $$C_m(\Tr(H_{l_1}),\ldots,\Tr(H_{l_m}))= N^{2-m}\Phi_N(l_1,\ldots,l_m)\to 0,$$ 
 as $N\to \infty.$
 The following theorem follows.
 \begin{theom}  Under $YM_N$,  the random family $(\Tr(H_l)-\esp[\Tr(H_l)])_{l\in\Ld(\R^2)}$  converges  weakly   as $N\to \infty,$ towards a Gaussian field  $(\phi_l)_{l\in\Ld(\R^2)}$, such that for any $a,b\in \Ld(\R^2),$ $\cov(\phi_a,\phi_b)=\Phi(\{a,b\})$.  If $(l_n)_{n\ge 0}$ is a fixed sequence of loops in $\Ld(\R^2) $ that converges towards $l\in\Ld(\R^2)$, then  $\phi_{l_n}\to\phi_l$ in distribution. If $h$ is an area preserving diffeomorphism of $\R^2,$  the process $(\phi_{h(l)})_{l\in\Ld(\R^2)}$ has the same law as $(\phi_l)_{l\in\Ld(\R^2)}$.
 \end{theom}
It is very tempting to apply  the result of section  \ref{Section Free energies BM}to study the  Yang-Mills measure with a potential. One can obtain   an improvement of Proposition \ref{UNIFOr} thanks to Lemma \ref{Bound second order}. Though, at this point,  arguing as in Theorem \ref{ConvCMSUP},  the present form \ref{Bound second order} would only allow to study a very restricted class loops and potentials.   

%Let us denote by $\Fc_1$ the set of function on $\Ld(\R^2)$ with countable support such that 
%$$\|V\|_1=\sum_{l\in \Ld(\R^2)} |V(l)|<\infty.  $$
%For any $V\in \Fc_1,$  under $YM_N$, almost surely the  following sum converge absolutly in operator norm and defines a random variable 
%$$V_N=\sum_{l\in \Ld(\R^2)}V(l) H_l. $$
%For any function $V\in \C^{\Ld(\R^2)},$ let us say that $V$ is symmetric if for any $l\in \Ld(\R^2),$ $V(l^{-1})=\overline{V(l)}.$ For any symmetric function $V\in\Fc_{1}$, the random matrix $V_N$ is hermitian and its operator norm is bounded by $\|V\|_1.$ In particular,  $0<\esp[e^{N\Tr(V_N)}]<\infty$. Let   $\YM_{N,V}$ be the probability measure on $(\Mca(\Ld(\R^2), \Un(N)),\Cc)$, whose density  with respect to $YM_N$ is $\esp[e^{N\Tr(V_{t,N})}]^{-1}e^{N\Tr(V_N)}.$  We shall  denote by $(H^V_l)_{l\in \Ld(\R^2)}$  the canonical process on $\Mca(\Ld(\R^2),\Un(N))$ with law $YM_{N,V}.$ 
%For any $V,W\in  \C^{W_q},$ let us define  for any $t\in \R_+^q,$
%$$\Jc_{t,V}(W)= \sum_{m\ge 1, a,w_1,\ldots, w_m\in W_q,T\in \Cc_m} |W(a)|\frac{\prod_{i=1}^m|V(w_i)|}{m!}T_t(w,w_1,\ldots, w_m)$$
%and 
%\begin{align*}
%\Jc_{t,2,V}(W)=&\sum_{a,b\in W_q} \la \overline{n}_a,\overline{n}_b\ra_t |W(a)||W(b)|\\& +\sum_{m\ge 1, a,b,w_1,\ldots, w_m\in W_q,T\in \Cc_m} |W(a)||W(b)|\frac{\prod_{i=1}^m|V(w_i)|}{m!}T_t(a,b,w_1,\ldots, w_m).
%\end{align*}
%For a fixed $V\in \C^{W_q}, t\in \R_+^q,$  we consider $\Fc_{t,V}=\{ W\in \Fc_{1,q}: \Jc_{t,1,V},\Jc_{t,2,V}<\infty    \}.$





\subsection{Small area limit \label{Section Small area limit}}


  
  
  
  
For any $\alpha>0$ and any loop $l\in \Ld(\R^2)$, denote by $\alpha.l$ the image of $l$ by the  dilatation of rate $\alpha,$  centered at $0$. If $\Sc=\{l_1,\ldots,l_m\}$ is a skein, $\alpha.\Sc= \{\alpha.l_1,\ldots,\alpha.l_m\}$. The following proposition shows that, as $\alpha\to 0$, all the quantities defined above have the same behavior as $\alpha\to 0$.

\begin{prop}
The following Taylor expansions are true for any $N\in\N^*$.   As $\alpha\to0, $ for any   loop $l\in\Ld(\R^2)$,
 $$\Phi_N(\sqrt{\alpha}. l)=1-\frac{\alpha}{2}\int_{\R^2} n_l^2(x)dx+O(\alpha^2)= \Phi(\sqrt{\alpha}. l)+ O(\alpha^2)$$
and for any  skein $\Sc$  with at least two loops,
\begin{align*}
\Phi_N(\sqrt{\alpha}.\Sc)&= (-\alpha)^{\#\Sc-1} \sum_{T_\Sc} \prod_{\{l_1,l_2\}\in T_\Sc}\int_{\R^2}n_{l_1}(x)n_{l_2}(x)dx+O(\alpha^{\#\Sc})\\
&=\Phi(\sqrt{\alpha}.\Sc)+O(\alpha^{\#\Sc}),
\end{align*}
where the sum is over connected graph with vertices $\Sc$ and $\#\Sc-1$ edges. In both cases, there exists a  positive  continuous function $b$, independant of $N$, such that $O(\alpha^{|\Sc|})\le \alpha^{|\Sc|}b(\sum_{l\in \Sc} \ell(l))$.
\end{prop}
\begin{dem} If $\Sc\in \Ec_\Ac,$ the assertion is a direct consequence of  Proposition \ref{Cumulant's control}. Continuity of the functions $\Phi_N,\Phi$ and $b$  allows then to conclude.\end{dem}

A direct consequence is the following
\begin{corol}  Let $W$ be a white noise on $\R^2$, with intensity given by the Lebesgue measure. As $t\to 0,$ the Gaussian field $(\frac{1}{t}\phi_{t. l})_{l\in \Ld(\R^2)}$, as well as, for any $N\in\N^*$, the random family $(t^{-1}(\Tr(H_{t.l})-N\Phi(t.l) ))_{l\in \Ld(\R^2)}$, under $YM_N,$  converge in distribution towards  the gaussian field $(i W(n_l))_{l\in \Ld(\R^2)}.$
\end{corol}




\section{Makeenko-Migdal equations}

We shall now address the problem of the  computation and characterization of the master field.  We say that $\Sc\in\Sk(\R^2)$ is a \emph{regular skein} if  its elements are smooth loops with transverse intersection of order $2$,  denote by $\Sk_{r}(\R^2)$ the set of regular skeins and by $\overline{\Sk_r}(\R^2)$ its quotient under diffeomorphisms.  For any integer $n$, the set of equivalence classes of skeins  with less than $n$ intersections is finite. Thanks to its invariance property under area-preserving diffeomorphisms and to its continuity, the master field  is  characterized by its value on $\Sk_r(\R^2)$ and yields functions  indexed by $\overline{\Sk_r}(\R^2)$, that we  show how to compute it inductively.




\subsection{Makeenko-Migdal equations for the master field on skeins}

For any skein $\Sc$, let us denote by $W_\Sc$ the expectation $\esp_{YM_N}\left[\prod_{l\in \Sc}\Tr(H_l)\right]$,  we call this function a \emph{Wilson skein}\footnote{We warn the reader that these functions are not normalized as they can be in the literature, so that, with this conventions, $W_{\text{cst}}=N$.} and say it is regular whenever the associated skein is. In view of the definition of discrete Yang-Mills measure, one may try to compute  the master field of higher order  of a  regular skein $\Sc$ using It� formula to yield a first order differential system for the family $(W_\Sc)_{\Sc\in \Sk_r(\R^2)}$, with  areas of the faces of a graph $\Gbb$ containing $\Sc$ as variables.  However, this differential system yields at first sight non-regular Wilson skeins $W_\Sc$ on $\Mca(\Pd(\Gbb), \Un(N))$, as features  the  example \ref{exampleInstability}.   \begin{example}  \label{exampleInstability}Consider a loop  $l$ that winds three times around the origin.  Let us name the faces $A,B$ and $C$ and choose a lassos basis $(l_A,l_B,l_C)$ according to a spanning tree as illustrated in figure \ref{SystemeInstableD3} in dashed lines.  In this basis, the loop is decomposed as $l=l_Cl_Bl_A^2l_Bl_A.$


\begin{figure}[!h] 
  \centering
 \includegraphics[height=1.5in]{SystemeInstableD3}\caption{ \label{SystemeInstableD3}}
\end{figure} 
Using It� formula as described in Lemma \ref{EDPTens} and differentiating with respect to the area of the faces $C$ and $B$ yields $\frac{d}{d|C|}W_l= -\frac{W_l}{2}$ and 
$$N\left(\frac{d}{d|B|}(W_l)+W_l\right)= -W_{l_Bl_A^2}W_{l_Cl_Bl_A}= -W_{l_A l_Bl_A}W_{l_Cl_Bl_A}.$$
These first two derivatives can be expressed in terms of regular Wilson skeins. However, the derivative with respect to the face of index 3 yields terms that do not seem to be polynomials of regular Wilson skeins: 
$$N\left(\frac{d}{d|C|}\left(W_{l}\right)+\frac{3}{2}W_l\right)= -W_{l_Bl_A}W_{l_Cl_Bl_A^2}-W_{l_Cl_Bl_A}^2-W_{l_A}W_{l_Bl_Al_Cl_Bl_A}. $$

\end{example}
For any regular skein $\Sc$, one must therefore face the problem of finding a closed system of Wilson skeins containing $W_\Sc$. The system of equations given by Lemma \ref{EDPTens} gives such a system, but its size happens to grow exponentially with the number of faces of the original skein (see section 6.8 of \cite{MF}, where the smallest closed system obtained is made of what is called Wilson garland). The Makeenko-Migdal equation solves this problem and  gives linear combinations of  area derivatives  operators that preserve the set of function indexed by  skeins, so that the size of the system grows as a polynomial in the number of faces.   Let $\Sc$ be a skein and $x$ be a point of intersection of its elements (between themself or each other).   Let us denotes by $\Sc_x$ the skein composed with the same loops as $\Sc $ except for the loop or the pair of loops  containing $x$ that is  replaced respectively  by the pair of loops or the  loop that instead of going straight along the same strand of $\Sc$, turns  at the point $x$ using the other strand with the same orientation (see figure \ref{lateraldodge}).

 \begin{figure} \centering \includegraphics[height=2 in]{LateralDodge}\caption{ \label{lateraldodge} Local transformation at an intersection point $x$ of a skein $\Sc$.}\end{figure}
The following proposition is proved in \cite{MF}, in a more  general framework and relies on integration by  parts applied to the product of  a function on $\Mca(\Pd(\Gbb),\Un(N))$ with the density  of the discrete Yang-Mills measure. We provide here another proof relying on the decomposition in lassos described in section \ref{Basis RL}  and on the invariance of the Brownian motion by adjunction. Let us fix a regular skein $\Sc$ and an embedded graph $\Gbb$ such that elements of $\Sc$ belongs to $\Pd(\Gbb)$.
\begin{prop}[Makeenko-Migdal equation] \label{Makeenko} Let $F_1,\ldots, F_4$ be the four faces of $\Gbb$ around a point of intersection  $x\in\Vbb$ of $\Sc$  in a cyclic order and such that $F_1$ is the face bounded by  the two edges of $\Sc$ leaving $x$. Then, 

\begin{equation}
\left(\frac{d}{d|F_1|}-\frac{d}{d|F_2|}+\frac{d}{d|F_3|}-\frac{d}{d|F_4|}\right)\esp_{YM_N}\left[\prod_{l\in \Sc}\Tr(H_l)\right]=\frac{1}{N}\esp_{YM_N}\left[\prod_{l\in \Sc_x}\Tr(H_l)\right].\label{Makequation}
\end{equation}
\end{prop}
For any skein $\Sc$, let us set $n_\Sc=\sum_{l\in\Sc}n_l$.  Notice that for $N=1$, the equality (\ref{Makequation}) is equivalent to the fact $$n_\Sc(F_1)^2-n_\Sc(F_2)^2+n_\Sc(F_3)^2-n_\Sc(F_4)^2=-2.$$ 
\begin {dem}   Let us denote  by $\mu_x$ the operator  $\frac{d}{d|F_1|}-\frac{d}{d|F_2|}+\frac{d}{d|F_3|}-\frac{d}{d|F_4|},$ where the faces are numbered as in the Proposition.  The strategy of our proof is to choose a spanning tree of  $\Gbb$ to decompose the  loops of the skein as described in section \ref{Basis RL} and to use Lemma \ref{EDP cut and join}  in such a way that terms on the left-hand-side of (\ref{Makequation}) cancel themselves so that a single cut and join transformation contributes.   

\vspace{0,2 cm}

\noindent\textbf{A differential equation indexed by skeins:} Let us fix a vertex $v\in \Vbb$, a spanning tree $T$ of $\Gbb$ and choose the lassos basis  $\Lambda$ of $\RL_v(\Gbb)$ associated to $T$ and $v$, as in section \ref{Basis RL}.  Decompose  each loop $l\in \Sc$  as in (\ref{decompositionenbeta}) and use the inversion formula (\ref{lassobeta}) \footnote{Note that the words obtained  may be non reduced,  for exemple, if $A$ is the only child of $B$ and is a leaf in $\hat{T}$, then  this decomposition is $\lambda_B =\beta_{e(B)}\beta_{e(A)}^{-1}= \lambda_B\lambda_A\lambda_A^{-1} $.} as is done in the example \ref{exampleInstability} for a single loop.  Let us order arbitrarily the elements of $\Sc$ and denote by   $S=(w_{l_1},\ldots ,w _{l_m})$  the family of words in lassos associated to element of $\Sc$, set $w=w_{l_1}\ldots w_{l_m}$ and $n=\ell(w).$ Let us denote by $t\in \R_+^{\Fbb},$ the vector of bounded faces area. According to the definition of the discrete Yang-Mills measure given in section \ref{Section Discrete YM} and recalling the notation of section \ref{Section differential system},  
$$W_\Sc= K_t(S,1_{\# \Sc}). $$
 By assumption, each edge $e\in\Ebb^+$ is used at most once by a loop of $\Sc$, we set $\ep_e=-1, $ if a loop of $\Sc$ goes backwards along $e$ and $\ep(e)=1, $ otherwise (in the example \ref{exampleInstability}, $\ep_e=1$  for any $e\in \Ebb^+$).  Therefore, in the decompositions of  $w_{l_1},\ldots,w_{l_m}$, each letter corresponds to a face $F$ and an edge $e\in [F,F_\infty]_{\hat{T}}$ visited by a loop of $\Sc$. Let us denote by  $\Gamma_T$ the set $\{(e,F)\in \left(\Ebb^+\setminus T^+\right)\times \Fbb: e\in [F,F_\infty] \text{ visited by }\Sc \}$. The ordering  of the elements of $\Sc$ and their decomposition in the basis $\Lambda$ yield an order $\lec$ on $\Gamma_T:$   $$w=\overset{\rightarrow}{\prod}_{(e,F)\in \Gamma_T}\lambda_F^{\ep(e)},  $$where the product is from left to right according to $\lec.$  The element  $(e,F)\in \Gamma_T$ are ordered with a lexicographic order $\lec$ taking into  account first the number of the loop visiting $e$, second the time of visit by a loop of $\Sc$ and thirdly the order $\lec_e$ on vertices of $\hat{T}_{e(F)}$ induced by the clockwise contour if $\ep(e)=1$ and its symmetric if $\ep(e)=-1$.    For any skein $\Sc'$, whose elements are obtain by permutation of the letters of elements $\Sc$ and any  two distinct elements $a,b\in\Gamma_T,$ with $a=(e_1,F)$, $b=(e_2,F)$, let us denote by $\Tc^{\ep(e_1)\ep(e_2)}_{a,b}(\Sc)$ or simply by $\Tc_{a,b}(\Sc)$ the cut and join transformation  of $\Sc$, as defined in section \ref{Section CutJoin}.    Then,  Lemma \ref{EDP cut and join} implies that 
\begin{equation}
\frac{d}{d|F|}W_\Sc=-\frac{\Lc_{[F,F_\infty]}(\Sc)}{2}W_\Sc -\frac{1}{N}\sum_{\substack{\hat{e},\hat{e'}\in [F,F_\infty]_{\hat{T} }\\ e\not=e'\text{ traversed by }\Sc  }}\ep(e)\ep(e')W_{\Tc_{(e,F),(e',F)}(\Sc)}.\label{difffacemot}
\end{equation}
%denote by $n$ the sum $\ell(w_1)+\cdots+\ell(w_m)$ and by $\sigma_\Sc\in\Sy_n$ the permutation $( 1\cdots \ell(w_{l_1}))\times\cdots\times ( 1\cdots \ell(w_{l_m}) ).$ Let $\{(U_{F,t})_{t\ge 0},F\in\Fbb\}$ be independent unitary Brownian motions. Then, 


\noindent\textbf{General compensations:}  Consider  two faces $A$ and $B$  such that $A$ is the first child of $B$ in $\hat{T}_{B}$ for $\lec_{e(B)}$.    Observe that  if  the letter $\lambda_B$ (resp. $\lambda_B^{-1}$) occurs   in the words $w_{l_1},\ldots,w_{l_m}$, then it is always on the left (resp. on the right) of $\lambda_A$ (resp. $\lambda_A^{-1}$). It follows that for any  pair of distinct edges $e_1,e_2$  such that $\hat{e}_1,\hat{e}_2\in [B,F_\infty]_{\hat{T}} $,   the reduction of the elements of $\Tc_{(e_1,A),(e_2,A)}^{\ep(e_1)\ep(e_2)}(\Sc)$ and $\Tc_{(e_1,B),(e_2,B)}^{\ep(e_1)\ep(e_2)}(\Sc)$ are equal. Therefore, equation (\ref{difffacemot})  implies that

\begin{equation}
\left(\frac{d}{d|A|}-\frac{d}{d|B|}\right) W_\Sc=-\frac{1}{2}W_{\Sc}-\frac{1}{N} \sum_{\substack{e' \in [A, F_\infty]_{\hat{T}}\\ \text{ traversed by }\Sc}} \ep(e(A))\ep(e') W_{\Tc_{(e(A),A),(e',A)}(\Sc)}. \label{pairedefaces}
\end{equation}
%\begin{align*}\Tr\left(\sigma_\Sc T_{\iota_{e_1,A},\iota_{e_2,A}} .w^\ts (U_{F,|F|},F\in\Fbb_b)\right)=&\Tr\left(T_{\iota_{e_1,A},\iota_{e_2,A}}.\sigma_\Sc  w^\ts (U_{F,|F|},F\in\Fbb_b)\right)\\
%=&\Tr\left(\sigma_\Sc T_{\iota_{e_1,B},\iota_{e_2,B}}.w^{\ts}(U_{F,|F|},F\in\Fbb_b)\right).
%\end{align*}


\vspace{0,5 cm}
 \noindent\textbf{Compensations for a specific spanning tree:} Consider now four faces satisfying the assumptions of the Proposition. Thanks to the �locality" of the relation  (\ref{Makequation}) and to the consistency of the discrete Yang-Mills measures, one may   further assume    that the  four faces around $x$ are distinct, not equal to $F_{\infty,\Gbb}$ and that they do not disconnect $\hat{\Gbb}$, that is,  the graph $\displaystyle(\hat{\Vbb}\setminus \{F_1,\ldots,F_4\},\Ebb\setminus \cup_{i=1}^4\{e, F_i\in e\})$ is connected.   Let us order the faces $F_1,\ldots,F_4$ counterclockwisely and  choose  a spanning tree $T$ of $\Gbb$ such that $F_2$ and $F_3$ are leaves of $\hat{T}$ and such that $(F_3,F_4,F_1)$ and $(F_2,F_1)$ are  paths of $\hat{T}$, as is displayed in figure \ref{PictureChoiceSpanningTree}.  
 \begin{figure}\centering\includegraphics[height=2 in]{ChoiceSpanningTree}\caption{A spanning tree of $\hat{\Gbb}$  that induces  an orientation on the edges around $x$ that matches the one induced by the  loops crossing at $x$ and  such that $F_2$ (resp. $F_3$) is the successor of $F_1$ (resp. $F_4$) for the orders  $\lec_e$, with  $e$ an edge such that $F_1,F_2\in \hat{T}_{F_L(e)}$ (resp. $F_3,F_4\in\hat{T}_{F_L(e)}$).\label{PictureChoiceSpanningTree}}\end{figure}
  Observe that the orientation of the three edges having $x$ as endpoints and belonging to $\hat{T}$, that is induced by the loops of $\Sc$, is the same as the one induced by such a  spanning tree. Moreover, $F_2$ (resp. $F_3$) is the successor of $F_1$ (resp. $F_4$) for the orders  $\lec_e$ with  $e$ an edge such that $F_1,F_2\in \hat{T}_{F_L(e)}$ (resp. $F_3,F_4\in\hat{T}_{F_L(e)}$).  One would  like now to apply twice formula (\ref{pairedefaces})   and compare $\left(\frac{d}{d|F_3|}-\frac{d}{d|F_4|}\right)W_{\Sc}$  with  $\left(\frac{d}{d|F_2|}-\frac{d}{d|F_1|}\right)W_{\Sc}$. Instead, let us remark the following.
  
  
  
  For any face $A,B\in \Fbb_b,$ let us denote by $\Theta_{A,B}$ the automorphism of the free group $\RL_v(\Gbb)$ that maps $\lambda_B$ to $\lambda_A\lambda_B\lambda_A^{-1}$ and fixes $\lambda_F$  for $F\not=B$.  According to Lemma \ref{Discrete YM}, for any skein $\Sc'$ with elements in $\Ld_v(\Gbb),$ 
$$W_{\Theta_{A,B}(\Sc')}=W_{\Sc'}.$$
The automorphism $\Theta_{A,B}$ transposes the letters  $\lambda_A$ and $\lambda_B$ in the decomposition of  $l_1,\ldots,l_m$ in $\Lambda.$  In particular, when  $A$ and $B$  are two faces such that $A$ is the first child of $B$ in $\hat{T}_{B}$ for $\lec_{e(B)}$   formula  \ref{pairedefaces} also applies with $\Theta_{A,B}(\Sc)$ in place of $\Sc.$  Notice that the letters  $\lambda_{F_2}$ and $\lambda_{F_3}$  are consecutives in the decomposition of elements $\Theta_{F_3,F_4}(\Sc)$ in $\Lambda$ (this is not true for $\Sc$) . Therefore, for any edge $e'\in [F_2,F_\infty]$ used by $\Sc$, 
\begin{equation}
\Tc_{(e(F_3),F_3),(e',F_3)}(\Theta_{F_3,F_4}(\Sc))= \Tc_{(e(F_2),F_2),(e',F_2)}(\Theta_{F_3,F_4}(\Sc)).\label{egalCompens}
\end{equation}
Now, applying twice formula  (\ref{pairedefaces})  to respectively  $\left(\frac{d}{d|F_3|}-\frac{d}{d|F_4|}\right)W_{\Theta_{F_3,F_4}(\Sc)}$  and  $\left(\frac{d}{d|F_2|}-\frac{d}{d|F_1|}\right)W_{\Theta_{F_3,F_4}(\Sc)}$  yields that 
   \begin{align}
N\mu_x .W_\Sc=&  \sum_{\substack{e' \in [F_2, F_\infty]_{\hat{T}}\\ \text{ traversed by }\Sc}} \ep(e(F_2))\ep(e') W_{\Tc_{(e(F_2),F_2),(e',F_2)}(\Theta_{F_3,F_4}(\Sc))} \label{Makmoinsun}\\
\nonumber&-  \sum_{\substack{e' \in [F_3, F_\infty]_{\hat{T}}\\ \text{ traversed by }\Sc}} \ep(e(F_3))\ep(e') W_{\Tc_{(e(F_3),F_3),(e',F_3)}(\Theta_{F_3,F_4}(\Sc))}. 
\end{align}
%$$T_{\iota_{e(F_3),F_3},\iota_{e',F_3}-\ep_{e'}}. \Theta_{F_3,F_4}(w)^{\ts}(U_{F,|F|},F\in \Fbb_b)=T_{\iota_{e(F_2),F_2},\iota_{e',F_2}}. \Theta_{F_3,F_4}(w)^{\ts}(U_{F,|F|},F\in \Fbb_b) .$$
The equalities (\ref{egalCompens}), together with  $\ep(e(F_3))=\ep(e(F_2))=1,$ yield that the equation (\ref{Makmoinsun}) gets simplified into
$$\mu_x. W_\Sc= \frac{1}{N} W_{\Tc_{(e(F_3),F_3),(e(F_4),F_3)}(\Theta_{F_3,F_4}(\Sc))}.$$
Let us observe that $\Tc_{(e(F_3),F_3),(e(F_4),F_3)}(\Theta_{F_3,F_4}(\Sc))=\Theta_{F_3,F_4}(\Sc_x), $  (see figure \ref{Cut Mak}  when $x$ is a  self-intersection point of an element of $\Sc$).
\begin{figure}\label{Cut Mak}\centering \includegraphics[height=3 in]{CutMak}\caption{  The single cut transformation contributing to Makeenko-Migdal equation, when $x$ is a self-intersection point of an element of $\Sc$. Here $e\in\Ebb$ denotes the edge with $\overline{e}=x,$ that does not belong to $T$.}
\end{figure}
Using once again the invariance of the discrete Yang-Mills measure under $\Theta_{F_3,F_4}$ implies the result.\end{dem}


If $x\in\Vbb$ is a point of intersection  point of one loop $l\in \Sc$, we denote  respectively  by  $\tilde{l}_x^L$  and $\tilde{l}_x^R$ the loop that turns left and  the loop that turns right at $x$. If $x$ is the intersection point of two different loops $l_1$ and $l_2$ of $\Sc$, we denote by $l_1\circ_xl_2 $ the concatenation of $l_1$ and $l_2$ at the point $x$. 

%For any partition $\pi$ of the skein $\Sc$, we set $$\Phi_N_\pi(\Sc)=\prod_{\Sc' \in \pi} \Phi_N(\Sc') .$$

\begin{theom}\label{coromak}Let $\Sc=\{l_1,\ldots, l_m\}$ be a regular skein, $x \in\Vbb_\Sc$ a point of intersection and  $F_1,\ldots, F_4$ faces around $x$ numbered as in Proposition \ref{Makeenko}. If $x$ is the intersection point of two different loops $l_1$ and $l_2$, 

\begin{equation*}
\left(\frac{d}{d|F_1|}-\frac{d}{d|F_2|}+\frac{d}{d|F_3|}-\frac{d}{d|F_4|}\right)\Phi_N( \Sc)=\Phi_N(l_1\circ_xl_2,l_3,\ldots, l_m).\tag{*}\label{Makun}
\end{equation*}
If $x$ is an intersection point of the loop $l_1$, then $\left(\frac{d}{d|F_1|}-\frac{d}{d|F_2|}+\frac{d}{d|F_3|}-\frac{d}{d|F_4|}\right)\Phi_N( \Sc)$ equals 
\begin{align*}\tag{**}\label{Makdeux}
\sum_{\substack{\Sc_x^L\sqcup\Sc_x^R=\Sc_x\\ \tilde{l}_{1,x,}^L\in \Sc_x^L \text{ and } \tilde{l}_{1,x}^R\in \Sc_x^R}}\Phi_N(\Sc_x^L)\Phi_N(\Sc_x^R)+\frac{1}{N^2}\Phi_N(\tilde{l}_x^R,\tilde{l}_x^L,l_2,\ldots, l_m).
\end{align*}
Moreover, if $F\in \Fbb_b$ is a neighbour face of the unbounded face, then 
\begin{equation*}
\frac{d}{d|F|}\Phi_N(\Sc)=-\frac{1}{2} \Phi_N(\Sc).\tag{***} \label{Makbord}
\end{equation*}

\end{theom}
\begin{dem} For any skein  $\{l_1,\ldots, l_m\}$ of $m$ loops and any $\pi,\Nu\in\Pc_m$ with $\pi\le \Nu$, let us  set  $\esp_\pi[l_1,\ldots,l_m]= \prod_{B\in\pi}\esp[\prod_{i\in B}\Tr(H_{l_i})]$, $C_\Nu[l_1,\ldots, l_m]= \prod_{B\in\Nu} C_{\#B}^N(\Tr(H_{l_i}),i\in B)$ and $C_{\pi,\Nu}=C_{\pi,\Nu}(\Tr(H_{l_i}),i \in \{1,\ldots,m\})$.   We shall consider the normalized cumulant
$$\Phi_\pi(l_1,\ldots,l_m)= \prod_{B\in \pi}\Phi(l_i, i\in B) = N^{m-2\#\pi}C_\pi(l_1,\ldots,l_m).$$ 
Assume that $x$ is an intersection point of $l_1$. For any partition $\Nu$ of $\{1,\ldots, m+1\}$  connecting $1$ with $2$,  denote by $\Nu'$ the partition of $\{1,\ldots,m\}$ obtained by identifying $1$ with $2$. Then for any partition $\Nu\in\Pc_{m+1}$ connecting $1$ with $2$, 


\begin{align*}
\sum_{1_{\{1,2\}}\le \pi\le \Nu} N\mu_x C_{\pi'}(l_1,l_2,\ldots,l_m)&=  \sum_{\pi\le \Nu'} N\mu_xC_\pi(l_1,l_2,\ldots,l_m)\\
&= N\mu_x \esp_{\Nu'}[l_1,l_2,\ldots,l_m]= \esp_{\Nu'}[\tilde{l}^L_{1,x}\tilde{l}^R_{1,x}, l_2,\ldots,l_m]\\
&=\esp_\Nu[\tilde{l}^L_{1,x},\tilde{l}^R_{1,x}, l_2,\ldots,l_m].
\end{align*}
Therefore,  for any partition $\pi\in \Pc_{m+1}$ connecting $1$ and $2$, 
$$C_{1_{\{1,2\}},\pi}(\tilde{l}^L_{1,x},\tilde{l}^R_{1,x},\ldots,l_m)= N\mu_x C_{\pi'}(l_1,l_2,\ldots,l_m).$$
In particular, 

\begin{align*}
\mu_x\Phi_N(l_1,\ldots,l_m)&= N^{m-3}C_{1_{\{1,2\}},1_{m+1}}(\tilde{l}^L_{1,x},\tilde{l}^R_{1,x},\ldots,l_m)\\
&=\sum_{\substack{\pi\in\Pc_{m+1}\\ \pi \vee 1_{\{1,2\}}=1_{m+1}}}N^{m-3}C_\pi(\tilde{l}^L_{1,x},\tilde{l}^R_{1,x},\ldots,l_m)\\
&=\sum_{\substack{\pi\in\Pc_{m+1}\\ \pi \vee 1_{\{1,2\}}=1_{m+1}}}N^{2|\pi|-4}\Phi_\pi(\tilde{l}^L_{1,x},\tilde{l}^R_{1,x},\ldots,l_m)
\end{align*}
If $\pi\in\Pc_{m+1}$ satisfies $\pi\vee 1_{\{1,2\}}= 1_{m+1}$, then, whether $|\pi|=1$ or $|\pi|=2$ and the equation (\ref{Makdeux}) follows. Assume now that $x$ is an  intersection point of $l_1$ with $l_2$. Then, for any partition $\pi\in\Pc_{m}$, $$\mu_x\esp_\pi(l_1,l_2,\ldots,l_m)=\left\{\begin{array}{ll} \frac{1}{N} \esp_\pi(l_1\circ_xl_2,\ldots,l_m) & \text{, if $1,2$  in the same block of $\pi$,}\\ 0 & \text{, otherwise.}  \end{array}\right.$$ Note that for any partition $\pi\in\Pc_m$, such that $1$ and $2$ are not in the same block of $\pi$, $\mu _xC_\pi(l_1,l_2,\ldots,l_m) =0. $
For any partition $\pi\in\Pc_{m-1},$ denote by $\tilde{\pi}\in \Pc_{m}$ the partition  obtained by shifting $\pi$ by $1$ and adding $1$ to the block containing $2$.  For any partition $\nu \in \Pc_{m-1}$, 

\begin{align*}
\sum_{\pi \le \nu}N\mu_x  C_{\tilde{\pi}} (l_1, l_2,l_3,\ldots, l_m)&= \sum_{\Wc\le \tilde{\nu}}N\mu_x  C_{\Wc} (l_1, l_2,l_3,\ldots, l_m)\\
&=N\mu_x\esp_{\tilde{\nu}}[l_1,l_2,\ldots,l_m]= \esp_\nu[l_1\circ_xl_2,l_3,\ldots,l_m].
\end{align*}
It follows that for any partition $\pi\in\Pc_{m-1}$, $$\mu_x C_{\tilde{\pi}}(l_1,l_2,\ldots,l_m)=\frac{1}{N}C_\pi(l_1\circ_xl_2,l_3,\ldots,l_m). $$
For $\pi=1_{m-1}$, the latter  equality yields  $$ \mu_x \Phi_N(l_1,l_2,\ldots,l_m)= \Phi_N(l_1\circ_xl_2,l_3,\ldots,l_m).$$

\end{dem}

Observe that equations (\ref{Makun}) and (\ref{Makbord})  on $\Phi_N$ do not depend on  $N$.



\subsection{Generalized Kazakov basis}
For any regular skein $\Sc$,    denote by  $\tilde{\Gbb}_\Sc=(\Vbb_\Sc,\Ebb_\Sc,\Fbb_\Sc)$ an embedded graph such that  $\Sc\subset\Pd(\tilde{\Gbb}_\Sc)$ minimizing $\#\Ebb_\Sc$ and $\#\Vbb_\Sc$.   We warn the Reader that   the image of edges of $\tilde{\Gbb}_\Sc$ are not necessarily contained in the image of loops of $\Sc$ and that $\tilde{\Gbb}_\Sc$ is not always unique. Consider  $\Gc_\Sc=(\Sc,E_\Sc)$ the graph with vertices indexed by $\Sc$, such that two loops are connected in $\Gc_\Sc$ if and only if they intersect each other. Then,  images of edges of $\tilde{\Gbb}_\Sc$ are included in the images of loops of $\Sc$ if and only if $\Gc_\Sc$ is connected.  In that case, $\tilde{\Gbb}_\Sc$ is the unique finest embedded graph such that $\Sc\subset\Pd(\Gbb_\Sc)$. In this  section, we shall fix a skein $\Sc$ such that $\Gc_\Sc$ is connected and set  $\Gbb=\tilde{\Gbb}_\Sc$. Let  $\Ebb^+$ and $\lambda$  be respectively  the orientation and the permutation of the edges $\Ebb$ induced by $\Sc$. We want now to determine whether  area-derivative operators can be obtained by linear combinations of   the operators appearing on the left-hand-side of Theorem \ref{coromak}.  To that purpose, let us set 
\begin{align*}
\mu:  \C^{\Fbb}&\longrightarrow \C^{\Ebb^+} \\
u&\longmapsto \left(e \mapsto u(F_L(e))-u(F_R(e))-u(F_L(\lambda^{-1}(e)))+u(F_R(\lambda^{-1}(e)))\right).
\end{align*}
For any loop $l\in \Ld(\Gbb)$, denote  respectively by $n_l\in \C^{\Fbb}$ and $\delta_l\in \C^{\Ebb^+}$ the winding number function of $l$ and the function $\displaystyle\sum_{l \text{�traverses } e}\delta_e$.  For any $v\in \Vbb,$ set $$*_v= \sum _{e\in \text{Out}(v)}\delta_e.$$ The following lemma is proved in \cite{MF}(Lemma 6.28.).
\begin{lemma}\label{maknoyau}i) The kernel of $\mu$ is spanned by $\{n_{l_1},n_{l_2},\ldots, n_{l_m},1_{\Fbb}\}.$


ii)   The image of $\mu$ is the orthogonal space to $\{*_v,v\in\Vbb\}\cup \{\delta_l,l\in\Sc\}.$


iii) The intersection of linear spaces spanned by $\{*_v,v\in\Vbb\}$ and $\{\delta_l,l\in\Sc\}$ is $\C1_{\Ebb^+}$.
\end{lemma}
Lemma \ref{maknoyau} yields that $\dim( \Im(\mu))= \#\Fbb-m-1=\# \Fbb_b-m.$  We have a first answer to our question: for any regular skein $\Sc$, the vector space spanned by the operators  $\{\mu(\nabla_\af)(e),e\in \Ebb^+\}$ is not $\text{span}(\{\frac{d}{d|F|}, F\in \Fbb_b\}).$
Nonetheless, for some skeins, the third condition  (\ref{Makbord}) allows to   complete the lacking information. 

 \begin{lemma}  \label{basediff}Suppose that there exists $m$ distinct faces $F_1,\ldots,F_m$  of $\Gbb$, neighbours of the unbounded face such that for any $i\in \{1,\ldots,m\}$, $l_i$ is bounding $F_i$. Let $\Fbb_{\infty,1}=\{F_\infty,F_1,\ldots,F_m\}$. Then, the application
\begin{align*}  
\overline{\mu}:  \C^{\Fbb}&\longrightarrow \Im(\mu)\oplus \C^{\Fbb_{\infty,1}}\\
u&\longmapsto (\mu(u),u_{| \Fbb_{\infty,1}})
\end{align*}
is an isomorphism.
\end{lemma}

\begin{dem} Thanks to Lemma \ref{maknoyau}, the source and the target of $\mu$ have the same dimension. If $u\in \ker(\mu)$, then there exists $\alpha\in \C^{m+1}$ such that  $u= \alpha_{n+1}1_{\Fbb}+\sum_{i=1}^m \alpha_i n_{l_i}$. Moreover, $u(F_\infty)=\alpha_{n+1}$ and for any $i\in\{1,\ldots, m\}$ $u(F_{i})=\alpha_i$. It follows that $\ker(\overline{\mu})=\{0\}$.
\end{dem}


We call a skein satisfying the condition of the Lemma a \emph{skein based at infinity}. Note that each vertex of $\Gbb$ has degree $4$, hence $\#\Ebb=2\#\Vbb$ and Euler relation implies $\#\Fbb=\#\Vbb+2$.  Therefore $\dim( \Im(\mu))=\#\Vbb-m+1.$ For any vertex  $v\in\Vbb$, let $e_1(v)$ and $e_2(v)$ be the two outgoing edges at $v$ ordered  clockwise and set $$\alpha_v=\delta_{e_1(v)}-\delta_{e_2(v)}\in\C^{\Ebb^+}.$$ 
Let us denote respectively  by $\Vbb_s$ and $\Vbb_f$ the set of self-intersection points of each loops  and the set of intersection points of pair of distinct loops  of $\Sc$. The type of crossing induces on $\Vbb_f$  an equivalence relation $\sim$ such two points $x$ and  $y$ are equivalent if they belong to the same loops. For any pair of  distinct points $x,y\in \Vbb_f$ such that $x\sim y$, denote by $\beta_{x,y}$ the function on $\Ebb^+$ that is $\alpha_x+\ep_{x,y}\alpha_y$, where $\ep_{x,y}=1$, if $e_1(x)$ and $e_1(y)$ belong to  different loops and $-1$ otherwise.    


\begin{lemma} i)For any $v\in \Vbb_s$ and any pair of vertices $x,y\in \Vbb_f$ belonging to the  the same pair of  loops, $\alpha_v,\beta_{x,y}\in \Im(\mu)$.



ii) Let $T_f$ be a spanning acyclic directed subgraph  of the complete graph on $\Vbb_f$  such that connected components of $T_f$ are the equivance class of $\sim$ on $\Vbb_f$.    The family $\{\alpha_v:v\in \Vbb_s\}\cup\{\beta_{x,y}: (x,y)\in T_f\}$ is  a free family of $\Im(\mu)$ and is a basis if and only $\Gc_\Sc$ is a tree.\label{baseimageKazakov}
\end{lemma}

\begin{dem} i) Indeed, for any $v\in\Vbb_s$ and $x,y\in \Vbb_f$ belonging to the same two of loops,  the vectors $\alpha_v$ and $\beta_{x,y}$ are orthogonal to  $\{*_v, v\in \Vbb\}$ and $\{\delta_l, l\in \Sc\}$.


\noindent ii) It is easy to see that thanks to the acyclycity of the graph $T_f$, the family $\{\alpha_v:v\in \Vbb_s\}\cup\{\beta_{x,y}: (x,y)\in T_f\}$ is free. Moreover, its cardinality is $\#\Vbb-m'$, where $m'$ is the number of connected components of $T_f$. The latter are in bijection with edges of $\Gc_\Sc$. Recall  that $\Gc_\Sc$  is connected  so that $m'=m-1$ if and only if $\Gc_\Sc$ is a tree. To conclude, recall that  $\dim(\Im(\mu))=\#\Vbb-m+1$.  \end{dem}


If $\Sc$ is made of a pair of loops that intersect each other, then the conclusion of ii) of Lemma \ref{baseimageKazakov} trivially holds, we give an example  of such a basis in figure \ref{base image mu}. There is a choice of directed acyclic graph that makes the decomposition in the basis easier, namely in each equivalence class of $\Vbb_f$, choose a base point  and connect any  other point towards it. If  $R_\Sc$  is a  set of class representatives for $\sim$, for any  $v\in \Vbb_f$,  we denote by $\overline{v}$ the unique element of $R_\Sc$ equivalent to $v$. % For any pair of loops $l,l'$, we   choose the  spanning tree on  the corresponding equivalent class $\Vbb_{l,l'}$ of $\sim$,  given by a bijection $\psi_{l,l'}:\Vbb_{l,l'}\to [1, \#\Vbb_{l,l'}]\cap \N.$ We call $(\psi_{l,l'})_{(l,l')\in \Gc_\Sc}$ a numbering of $\Vbb_f$. A simpler choice: stars!
%In the example shown on figure \ref{base image mu}., the spanning tree $T_f$ on $\Vbb_f$ is traced in dashed line and the support of the elements of the family described in b) of Lemma \ref{baseimageKazakov} are disjoint, we represent the values on their elements on the edges of $\Gbb$, 
\begin{figure}[!h]   \centering 
 \includegraphics[height=2in]{Basepairelacetsaretes} 
 \caption{\label{base image mu}Here is the basis of $\Im(\mu)$ associated to the spanning tree on $\Vbb_f$ drawn with dashed lines. In this example, if the supports of two basis functions intersect, then  the two functions take the same value on this intersection that we print on the edge.}
 \end{figure}.
 \begin{lemma}Let   $R_\Sc$ be set of class representatives of $\sim$,  the family  $\{\alpha_v,v\in \Vbb_s\}\cup\{\beta_{v,\overline{v}},v\in \Vbb_f\setminus R_\Sc\}$ is a free family and for  any function $\vp$ belonging to its span,
  \begin{equation}\label{inversionKazakovarbre}
\vp= \sum_{v\in \Vbb_s} \varphi(e_1(v))\alpha_v   +\sum_{v\in \Vbb_f\setminus R_\Sc}\vp(e_1(v))\beta_{v,\overline{v}}. 
\end{equation}
 \end{lemma}
% \sum_{(l,l')\in E_\Sc, i < \#\Vbb_{l,l'}}\alpha_{\psi_{l,l'}(i,i+1)}\beta_{\psi_{l,l'}(i,i+1)}, with $$\alpha_{\psi_{l,l'}(i,i+1)}=\vp\left(e_1(\psi_{l,l'}(i))\right)-\sum_{j<i}\left(\prod_{k=1}^{j} \ep_{\psi_{l,l'}(k,k+1)}\right)\vp(e_1(\psi_{l,l'}(j))).$$
 
These free-families have the following pre-image under the function $\mu.$ For any $v\in \Vbb_s$, let $l_v$ be the loop that starts with $e_1(v)$ and stops at its first return at $v$. For any pair  $(x,y)$ of distinct vertices in $\Vbb_f$ that are intersection of the same pair of loops, let $l_{x,y}$ be the loop that starts with $e_1(x)$, uses the same loop until it reaches $y$ and  then goes back to $x$ using the second loop. See figure \ref{baselacets}, where we draw the pre-image of the family described in figure \ref{base image mu}.
\begin{lemma}  For any $v\in \Vbb_s$  and any pair $(x,y)$  of distinct equivalent vertices of $\Vbb_f$,\begin{equation}
\mu(n_{l_v})=\alpha_v
\end{equation}
and 
\begin{equation}
\mu(n_{l_{x,y}})=\beta_{x,y}.
\end{equation}
\end{lemma}
 \begin{figure}[!h] 
  \centering
 \includegraphics[height=2in]{Basepairelacets}\caption{ \label{baselacets}The application  $\mu$  maps the winding number functions of loops drawn on this figure to the basis represented in figure \ref{base image mu}. These family of winding number functions  completed by the winding number functions of elements of $\Sc$ is a basis of $\C^{\Fbb}$.}
\end{figure} 

%If $\Gc_\Sc$ have cycles, we shall complete the  basis given by Lemma \ref{baseimageKazakov}  in the following way. Let us fix a spanning tree $T_\Sc$ of $\Gc_\Sc$ and an acyclic graph $T_f$ as in Lemma \ref{baseimageKazakov}. Fix further, a family $\Ic$ of points $\{x_{(l,l')}\in\Vbb_f:(l,l')\in \Gc_\Sc\}$  such that for any $(l,l')\in E_\Sc$, $x_{(l,l')}\in l\cap l'$ and   $x_{(l,l')}$ is an endpoint of the directed graph $T_f$. Denote further by $\Ic^*$ the family of starting points of $T_f$ that is paired with $\Ic$.



\vspace{0,5 cm }

If $\Gc_\Sc$ have cycles, we shall complete\footnote{Note that such considerations are not useful to compute the correlation functions of the Gaussian master field thanks to Makeenko-Migdal relations.} the  basis given by Lemma \ref{baseimageKazakov}  in the following way. For each edge $(l_1,l_2)\in E_\Sc$, we denote by $v_{l_1,l_2}$ the element of $R_\Sc$ at the intersections of $l_1$ and $l_2$.   For each edge $(l,l')\in E_\Sc \setminus T_\Sc$,  consider    the unique path $(l_1,l_2,\ldots, l_{k})$  in $T_\Sc$ from $l$ to $l'$ and  for any $i\in \{1,\ldots, k\}$,  let $c_i$  be the regular path that is the restriction of $l_{i+1}$ such that  $\underline{c_i}=v_{l_i,l_{i+1}}$,  $\overline{c_i}=v_{l_{i+1},l_{i+2}}$ (where loops are indexed by $\Z \slash k\Z$).  The concatenation $c_1c_2\ldots c_k$ is a loop of $\Gbb$, that follows elements of the cycle $(l_1,l_2,\ldots,l_k,l_1)$ and  change from one strand to another at the points of $R_\Sc$. The application $\mu$ maps the loop $\tilde{l}_{l,l'}$ to  

$$\gamma_{l,l'}=\mu(\tilde{l}_{l,l'})= \epsilon_1\alpha_{v_{l_1,l_2}}+\ep_{2}\alpha_{v_{l_2,l_3}}+\ldots+\ep_k\alpha_{v_{l_k,l_1}},$$
where for any $i\in\Z\slash k\Z$,  $\epsilon_i=1,$ if $e_1(x_i)\in l_{i+1}$ and $-1$ if $e_1(x_i)\in l_{i}$ (with the above notation,  $\ep_i= \ep_{v_{l_{i-1},l_{i}},v_{l_{i},l_{i+1}}}$). Let us write $\ep_{l',l}=\epsilon_k$.  

\begin{lemma}The family $\{\alpha_v:v\in \Vbb_s\}\cup\{\beta_{v,\overline{v}}: v\in \Vbb_f\setminus R_\Sc\}\cup\{\gamma_{l,l'} : (l,l')\in E_\Sc\setminus T_\Sc\}$ is  a basis of $\Im(\mu)$. Moreover for any function $\varphi\in \C^{\Ebb^+}$ belonging to $\Im(\mu)$, 


\begin{align*}
\varphi = \sum_{v\in \Vbb_s} \varphi(e_1(v))&\alpha_v+\sum_{v\in \Vbb_f\setminus R_\Sc}\varphi(e_1(v))\beta_{v,\overline{v}}\label{inversionKazakov}\tag{K}\\
&+ \sum_{(l,l')\in E_\Sc\setminus T_\Sc}\ep_{l,l'}\left(\varphi(e_1(\overline{v_{l,l'}}))-\sum_{\overline{v}=v_{l,l'}}\ep_{v,v_{l,l'}}\varphi(e_1(v))\right)\gamma_{l,l'}.
\end{align*}


\end{lemma}
\begin{dem} First, we should notice that $\#R_\Sc = \# E_\Sc=m-1+\#E_\Sc\setminus T_\Sc $ and that this family has the good cardinality: 
 $$\#\Vbb_s+\left(\#\Vbb_f-(m-1) -  \#(E_\Sc\setminus T_\Sc)\right)+ \#(E_\Sc\setminus T_\Sc)=\#\Vbb-m+1=\dim(\Im(\mu)).$$ To conclude, we check that the relation (\ref{inversionKazakov}) holds true for any  function $\vp$ in the span of this family. We denote by $R'$ the subset of $R_\Sc$  representing classes indexed by $E_\Sc\setminus T_\Sc. $ Let $c\in \C^{\Vbb\setminus R_\Sc\cup R'}$ be  a vector such that  $\vp= \sum_{v\in \Vbb_s} c_v\alpha_v+\sum_{v\in \Vbb_f\setminus R_\Sc}c_v\beta_{v,\overline{v}}+ \sum_{(l,l')\in E_\Sc\setminus T_\Sc}c_{v_{l,l'}}\gamma_{l,l'}$. Let $r$ belong to $ R'$ and let $(l,l')$ be the edge $ E_\Sc\setminus T_\Sc$ such  that  $r\in l\cap l'$. Then,   $\gamma_{l,l'}(e_1(r))=\ep_{l',l} $ and  for any $v\in \Vbb_f\setminus R_\Sc$, such that $\overline{v}= r,$  $\beta_{v,r}(e_1(r))= \ep_{v,r}$, whereas  $e_1(r)$ cancels any other element of the family.  What is more,  for any $v\in \Vbb\setminus R_\Sc,$  $\gamma_{l,l'}(e_1(v))=0$.   This computation immediatly yields the formula (\ref{inversionKazakov}).
  \end{dem}









%loop that starts at $x_1$ following $l_2$, until it reaches $x_2$, then uses $l_3$, until it reaches $x_3$ 


%that is the concatenation of the restrictions of $l_{i+1}$ between the points $x_i$ and $x_{i+1}$. 
% Let $T_f$ be an acyclic graph with vertices $\Vbb_f$ as in Lemma \ref{baseimageKazakov}. 
%that is the intersection of two loops $l_i$ and $l_j$ with $i<j$,  leaving $v$ along $e_i$ and $e_j$,  set $$\alpha_v= \delta_{e_i}-\delta_{e_j}.$$
Let us give the simplest example, where $\Gc_\Sc$ is not a tree, that is when $\Sc$ is made of three cycles that intersect each others twice. The graph $\Gc_\Sc$ is a triangle, we choose as set of representatives the three points of intersections lying on the boundary of $F_\infty$. We draw on figure the families of loops one get when the circles have the same orientation.
 \begin{figure}[!h] 
  \centering
 \includegraphics[height=3in]{TroisCercles}\caption{ \label{TroisCercles} Here is the loops that are mapped to the basis $\beta$ and $\gamma$, for three circles counterclockwisely oriented. The class representative of $\sim$ are taken on the boundary of $F_\infty$. In this case, the loop $\tilde{l}_{l,l'}$ does not depend on the choice of the spanning tree of $\Gc_\Sc$ and is drawn with dashed lines, whereas the familly $\beta$ has three elements. }
\end{figure} 





Let us see how the above construction answer our initial question. For any loop $l\in \Sc$, $\overline{\mu}(n_l)= \epsilon_l 1_{F_l}\in\C^{\Fbb_{\infty,1}}$, where $\epsilon_l=n_l(F_l)\in\{-1,1\}$. Hence, for any set of class representatives $R_\Sc$ for $\sim$ and $T_\Sc$ a spanning tree of $\Gc_\Sc$, the family $\{n_{l_{v,\overline{v}}} , v\in \Vbb_\Sc\setminus R_\Sc \}\cup\{n_{l_v},v\in\Vbb_s\}\cup \{n_{\tilde{l}_{l,l'}}:(l,l')\in E_\Sc\setminus T_\Sc\} \cup\Sc\cup\{1_\Fbb\}$ is a basis of $\C^{\Fbb}$ and its   image under $\overline{\mu}$ is the free family $\{\alpha_v, v\in \Vbb_s\}\cup\{\beta_{x,y},(x,y)\in T_f\}\cup \{\gamma_{l,l'},(l,l')\in E_\Sc\setminus T_\Sc\}$ completed by the canonical basis of $\C^{\Fbb_{\infty,1}}$. Moreover, for any  $\varphi\in \Im(\mu)\oplus\C^{\Fbb_{\infty,1}}\subset \C^{\Ebb^+}\oplus\C^{\Fbb_{\infty,1}}$, 

\begin{align*} \label{decompositionbaseimage}
\varphi = \sum_{v\in \Vbb_s}\varphi(e_1(v))\alpha_v&+\sum_{v\in \Vbb_f\setminus R_\Sc} \varphi(e_1(v))\beta_{x,y}+ \sum_{F\in \Fbb_{\infty,1}}\varphi(F_l)\delta_F  \tag{$\star$}\\
& + \sum_{(l,l')\in E_\Sc\setminus T_\Sc}\ep_{l,l'}\left(\varphi(e_1(v_{l,l'}))-\sum_{\overline{v}=v_{l,l'}}\ep_{v,v_{l,l'}}\varphi(e_1(v))\right)\gamma_{l,l'}
\end{align*} 
and 
\begin{align*}
\overline{\mu}^{-1}(\varphi)=\sum_{v\in \Vbb_s}\varphi&(e_1(v))n_v+\sum_{v\in \Vbb_f\setminus R_\Sc} \varphi(e_1(v))n_{v,\overline{v}}+ \sum_{l\in\Sc}\epsilon_l\varphi(F_l)n_l \\
&+ \sum_{(l,l')\in E_\Sc\setminus T_\Sc}\ep_{l,l'}\left(\varphi(e_1(v_{l,l'}))-\sum_{\substack{v\in\Vbb_f\\\overline{v}=v_{l,l'}}}\ep_{v,v_{l,l'}}\varphi(e_1(v))\right)n_{\tilde{l}_{l,l'}}.
\end{align*} 
If a skein is based at infinity, considering $\varphi= \overline{\mu}(\nabla_{\af})$ yields a complete answer to our initial question.    For any face $F\in\Fbb$,
%For any differentiable function $\Theta$  of the area s of the faces $\Fbb$,


\begin{align*}
\frac{d}{d|F|}& =  \sum_{v\in \Vbb_s}n_v(F)\mu(\nabla_\af)(e_1(v)) +\sum_{v\in \Vbb_f\setminus R_\Sc} n_{v,\overline{v}}(F)\mu(\nabla_\af)(e_1(v))+ \sum_{l\in\Sc}\epsilon_ln_l(F)\frac{d}{d|F_l|}  \\
&+ \sum_{(l,l')\in E_\Sc\setminus T_\Sc}\ep_{l,l'}n_{\tilde{l}_{l,l'}}(F)\left(\mu(\nabla_\af)(e_1(v_{l,l'}))-\sum_{\substack{v\in\Vbb_f\\\overline{v}=v_{l,l'}}}\ep_{v,v_{l,l'}}\mu(\nabla_\af)(e_1(v))\right).\tag{$\diamond$}
\end{align*}
%\left(\ep_{l,l'}\mu(\nabla_\a)(e_1(x_{l,l'}))-\mu_{\nabla_\a}(e_1(x_{l,l'}^*))\right).

Using Theorem \ref{coromak}, we have an expression for all  area derivatives of  $\Phi(\Sc)$, for any skein based at infinity and such that $\Gc_\Sc$ is connected. What is more, note that if $\Sc$ is based at infinity and $\Gc_\Sc$ is not connected, then $ \Phi_N(\Sc)=0. $     Nonetheless, observe that if $v\in\Vbb_s$, it may happen, as in example of figure \ref{baselacets} at the only point of $\Vbb_s$, that among the two skeins $\Sc_v^L$ and $\Sc_v^R$   obtained by splitting  $\Sc$ at $v$, one of them is  not based at infinity.   To solve this problem and compute the master field against  all skeins, we could enlarge the type of loops families. Instead, observe that any skein can be obtained by putting some areas of a skein based at infinity to zero. We must now prove that  these equations  characterize  the higher-order master field $\Phi$.
%\begin{example}
%\end{example}
%\begin{example}
%\end{example}


\subsection{Complexity of skeins}

We shall consider in this last section a slightly different notion of embedded graph. We call a \emph{multi-embedded graph in the plane} a triplet $\mathbb{G}=(\Vbb,\Ebb,\Fbb_b)$  satisfying the same conditions as an embedded graph as defined at the beginning of section \ref{section embedded graphs} but where the element of $\Fbb_b$ are allowed to be non-simply connected. 

\vspace{0,2 cm}


Let us fix a regular skein $\Sc$. We let    $\Gbb_\Sc=(\Vbb_\Sc,\Ebb_\Sc, \Fbb_\Sc)$ be  the finest multi-connected embedded graph such that $\Sc\subset\Pd(\Gbb_\Sc).$  The graph $\Gbb_\Sc$ is connected if and only if $\Gc_\Sc$ is connected.
%s a mult as  triple $(\Vbb_\Sc,\Ebb_\Sc,\Fbb_\Sc)$ such that $\Vbb\subset \R^2$, $\Ebb\subset \Pd(\R^2)$, and $\Fbb_\Sc$ is a finite set of open connected domain of $\R^2$ with piecewise smooth boundaries included in the image of $\Ebb$.    
For any $l\in \Pd(\Gbb_\Sc)$, we  consider   $$d_{\infty,\Sc}(l)= \inf\{d_{\hat{\Gbb}_{\Sc}}(F,F_\infty)-1: F\in \hat{\Gbb}_{\Sc}, F\cap F_{\infty,\Gbb_{\{l\}}}=\emptyset \}.$$   Let $I(\Sc)$ be the number of intersections of $\Sc$ and for any loop $l\in\Sc$ denote by $I_\Sc(l)$ the number of intersections of $l$ with itself and other loops of $\Sc$. We define the \emph{complexity} of $\Sc$ to be the number  


 $$\Cc(\Sc)= I(\Sc) + 2 \sum_{l\in\Sc} d_{\infty,\Sc}(l).$$ 
 \begin{example} A skein $\Sc$ is based at infinity if and only if $\Cc(\Sc)=I(\Sc).$ 
 \end{example}
 \begin{example}  If $\Cc(\Sc)=0$, then $\Sc$ is an union of closed Jordan curved bounding disjoints domains. Therefore, $\Phi(\Sc)=0,$ if $\#\Sc\ge 2$ and $e^{-\frac{|D|}{2}}$, if $\Sc$ has a single loop bounding a simply connected domain $D.$ \end{example}
 
 \begin{lemma} \label{deccomplex}\begin{itemize}\item[i)]For any point of intersection $v$ of a skein $\Sc$,  if $v\in \Vbb_f$, $$\Cc(\Sc_v)<\Cc(\Sc)$$
 
 and if $v\in\Vbb_s$, for any partition $\Sc_v^L\sqcup \Sc^R_v=\Sc_v$, $$\max\{\Cc(\Sc_v^L),\Cc(\Sc_v^R)\}<\Cc(\Sc).$$
 \item[ii)] For any regular skein $\Sc$, there exists a family  $(\Sc^\ep)_{\ep>0}$  of skeins based at infinity  with $\Cc(\Sc^\ep)=\Cc(\Sc)$ for any $\ep>0$ and a family of smooth paths $(r_l)_{l\in \Sc}$ such that  for any $l\in\Sc$, $\underline{r_l}\in l$ and $\Sc^\ep$ converges to  $\{r_l l r_l^{-1},l\in\Sc\}, $ as $\ep\to 0$.
 \end{itemize}
 \end{lemma}
 
 
 \begin{dem} i)  Assume that $v\in \Vbb_f$. Then,  for any loop $l\in \Sc$, that does not contain $v$, $l\in \Sc_v$ and $d_{\infty, \Sc_v}(l)\le d_{\infty,\Sc}(l)$. If $l_1$ and $l_2$ are the two loops crossing at $v$, then $d_{\infty, \Sc_v}(l_1\circ_vl_2)\le \min\{d_{\infty, \Sc}(l_1),d_{\infty, \Sc}(l_2) \}$.  Moreover, $\Vbb_s(\Sc_v)= \Vbb_s(\Sc)$ and for any  $w\in \Vbb_s(\Sc)$, $d_{\infty, \Sc_v}(l_v)\le d_{\infty, \Sc}(l_v)$. Therefore, the fact that $I(\Sc_v)=I(\Sc)-1$ yields  the expected inequality. 
 
 
 
 
 \noindent Assume now that  $v\in \Vbb_s$. Let $l\in\Sc$ be the loop of  $\Sc$ crossing at $v$. Recall  that  $l_{v,L}$ and $l_v^R$ are the loops that turns respectively to the left and to right at $v$ (so that they use respectively the edge $e_1(v)$ and $e_2(v)$).  Fix a partition $\Sc_v^L\sqcup \Sc_v^R$ of $\Sc_v$ separating these two loops.   For any loop $l'\in\Sc_v\setminus \{l_v^L,l_v^R\}$, $d_{\infty,\Sc^L_v}(l'),d_{\infty,\Sc^R_v}(l')\le d_{\infty,\Sc}(l')$.  Suppose that there exists a path $c\in \Pd(\hat{\Gbb}_{\Sc})$ such that  $ F_{\infty,\Gbb_{l^L_v}}\cap\underline{c}=\emptyset$ and $\ell(c)-1= d_{\infty,\Sc^L_v}(l_v)=d_{\infty,\Sc}(l)$.  Then, on the left side, $I(\Sc_v^L)\le I(\Sc_v)-1$ and  $\Cc(\Sc^L_v)\le \Cc(\Sc)-1$.  The right side needs more caution.  Consider the paths $c^{\pm}\in \Pd(\hat{\Gbb}_{\Sc_v})$ that start with $(F^R(e_1(v)),F^L(e_1(v)))$,  board  $l^L_v$   respectively on the right and on the left, such that one path follows the orientation of $l_{v,L}$ and the other goes in the reverse direction,  until they cross $c$ and then follow $c$ up to $F_{\infty,\Gbb_{\Sc_v}}$.  Their combinatorial length satisfies
 
 
  $$\ell(c^+)+\ell(c^-)\le 2+I_{\Sc_v}(l_v^L)+2\ell(c).$$
 Therefore, $$d_{\infty,\Sc^R}(l_v^R)\le  \min\{\ell(c^+),\ell(c^-)\}-1\le \frac{I_{\Sc_v}(l_v^L)}{2}+ d_{\infty,\Sc}(l).$$ The number of intersections of the right skein $\Sc_v^R$ is bounded by $I(\Sc\setminus\{l\})+I_{\Sc_v}(l_v^R)$. Moreover, for any loop $l'\in \Sc_v^R\setminus\{ l_v^R\}$, $d_{\infty,\Sc^R_v}(l')\le d_{\infty,\Sc}(l').$
 It follows that  $$\Cc(\Sc^R_v)\le  I(\Sc\setminus \{l\})+I_{\Sc_v}(l_v^R) +I_{\Sc_v}(l_{v,L})+  2\sum_{l'\in \Sc\setminus\{l\}} d_{\infty,\Sc}(l').$$
The equality   $I_{\Sc_v}(l_v^R) +I_{\Sc_v}(l_v^L)+ I(\Sc\setminus \{l\})=I(\Sc_v)= I(\Sc)-1$  concludes.

ii)  For each $l\in\Sc$, such that $d_{\infty,\Sc}(l)>0$, consider a self-avoiding path $c_l\in\Pc(\hat{\Gbb}_\Sc)$ such that $ F_{\infty, \Gbb_l}\cap \underline{c}_l=\emptyset$ and $\overline{c}_l =F_\infty(\Gbb_\Sc)$.  Choose such a family $(c_l)_{l\in \Sc}$   of loops that do not cross each other but may be merged with one another. Deform each loop $l$ along $c_l$  into $\tilde{l}$  so that the deformation intersects exactly twice each dual edge  of $c_l$ and does not intersect the deformation of other loops. Denote by $\tilde{\Sc}$ the skein  $\{\tilde{l}: l\in \Sc, d_{\infty,\Sc}(l)>0\}\cup\{l: l\in \Sc, d_{\infty,\Sc}(l)=0\}$.  By construction, for any $l\in\tilde{\Sc}$, $d_{\infty,\tilde{\Sc}}(l)=0$ and $$\Ic(\tilde{\Sc})=I(\Sc)+2 \sum_{l\in\Sc}(|c_l|-1)=\Cc(\Sc).$$
 Each path $c_l\in \Pd(\hat{\Gbb}_\Sc)$ induces  a path $\tilde{c}_l\in \Pd(\hat{\Gbb}_{\tilde{\Sc}})$ such that faces belonging to $\tilde{c}_l$ are boarded by $\tilde{l}$. When the areas $(|F|)_{F\in \cup_{l\in \Sc}c_l } $ go to zero, the vector of random variables $(H_{l})_{l\in\tilde{\Sc}}$ converges in distribution to $(H_{r_l l r_l^{-1}})$, where $ r_l$ are smooth paths such that $\underline{r_l}\in l$.  For any loop $l\in \Sc$, $H_{r_llr_l^{-1}}=H_{r_l}^{-1}H_lH_{r_l}$ and $\Tr(H_{r_llr_l^{-1}})=\Tr(H_{l})$. Considering $\Phi_N(\tilde{\Sc})$ and $\Phi_N(\Sc)$ as functions of $(|F|)_{F\in \Fbb_{\tilde{\Sc}}}$,  it follows that $$\Phi_N(\tilde{\Sc})\to \Phi^{N}(\Sc),$$ as $(|F|)_{F\in \Fbb_{\tilde{\Sc}}}\to 0.$  
 \end{dem}
 
 
 We can now solve our differential system recursively ordering skeins by their complexity. Recall that if $x$ is a point of intersection of a skein $\Sc$, then $\mu_x=\frac{d}{d|F_1|}-\frac{d}{d|F_2|}+\frac{d}{d|F_3|}-\frac{d}{d|F_4|}$, where $F_1,F_2,F_3$ and $F_4$ are faces around the vertex $v$ in cyclic order and $F_1$ is the face bounded by the two outgoing edges of $x$.  For any pair  of  skeins $\Sc$ and $\Sc'$, let us say that $\Sc$ and $\Sc'$ are equivalent and write $\Sc\sim\Sc'$ if there exists a family $\{c_l, l\in \Sc\}$ of paths of $\Pd(\R^2)$  such that $\underline{c_l}\in l$ for any $l$ and  $\Sc'=\{c_l lc_l^{-1}, l\in \Sc\}.$
%\renewcommand{\labelenumi}{(\Roman{enumi})}
 \begin{theom} \label{Thm caracterisation par MM} there exists a unique function $\Phi$ on  $\Sk(\R^2)$ satisfying the following equations. 
 \begin{enumerate}[1.]
 \item $\Phi(\{1\})=1.$
 \item If $\Sc^-$ and $\Sc^+$ are two skeins that are separated by a closed Jordan curve, $\Phi(\Sc^-\sqcup\Sc^+)=0$.
 \item $\Phi$ is continuous for the topology of $1$-variation.
 \item If $\Sc\sim \Sc'$, then $\Phi(\Sc')=\Phi(\Sc).$ \item  For any area-preserving  diffeomorphism $g$ of the plane, $\Phi\circ g =\Phi.$
 \item  For any regular skein  $\Sc$, $\Phi$ is differentiable with respect to $(|F|)_{F\in \Fbb_\Sc}$ and satisfy the following differential equations. If $x$ is the intersection of two different loops,
 \begin{equation}
\mu_x \Phi(\Sc) =\Phi(\Sc_x) .
\end{equation}
  If $x$ is the intersection of a loop of $\Sc$ with itself,
  \begin{equation}
\mu_x \Phi(\Sc)= \sum_{\substack{\Sc_x^L\sqcup\Sc_x^R=\Sc_x\\ \tilde{l}^L_{1,x}\in \Sc_x^L \text{ and } \tilde{l}^R_{1,x}\in \Sc_x^R}}\Phi_N(\Sc_x^L)\Phi_N(\Sc_x^R).
\end{equation}
For any face $F\in\Fbb_\Sc$, neighbour of $F_\infty$,
\begin{equation}
\frac{d}{d|F|} \Phi(\Sc) =-\frac{1}{2}\Phi(\Sc). 
\end{equation}
\end{enumerate}
 \end{theom} 
 
 \begin{dem} The function $\Phi_N$ satisfies by construction the point 1.-4.. For any regular skein $\Sc$, $\Phi_N(\Sc)$ is analytic in $(|F|)_{F\in\Fbb_\Sc}$, satisfies $(*)$, $(**)$ and $(***)$ and converges uniformly on every compact set of $\R_+^{\Fbb_\Sc}$ to the function $\Phi(\Sc)$. Therefore, $\Phi(\Sc)$ is analytic and satisfies the equations of point 5. Let us now consider the question of uniqueness. Let $\Psi$ be a function on finite skeins satisfying point 1 to 5.   Using points 3 and  4, it is enough to prove that $\Psi(\Sc)=\Phi(\Sc)$ for any regular skein $\Sc$. For any integer $n$, denote by $\mathsf{Sk}_n$ the set of regular skeins of complexity less than $n$. Let us prove inductively that $\Psi_{|\Sk_n}=\Phi_{|\Sk_n}$. Thanks to points 1 and 2, the equality holds for $n=0$. Assume that it is true for $n\in\N$ and consider a regular skein $\Sc\in\Sk_{n+1}$. Suppose that $\Sc$ is based at  infinity. According to $(\diamond)$ and to point 6, for any face $F\in\Fbb_\Sc,$ $\frac{d}{d|F|}\Phi$ and $\frac{d}{d|F|}\Psi$ are a linear combination of  terms of the form  $\Phi(\Sc'),\Psi(\Sc')$ or $\Phi(\Sc^L)\Phi(\Sc^R),\Psi(\Sc^L)\Psi(\Sc^R)$, with $\Cc(\Sc'),\Cc(\Sc^L),\Cc(\Sc^R)<n$. Hence, by induction hypothesis, $\Psi(\Sc)=\Phi(\Sc)$. Assume now that $\Sc$ is not based at infinity. Let $(\Sc^\ep)_{\ep>0}$ and $\{r_l, l\in\Sc\}$ be given as in Lemma  \ref{deccomplex}. Then, for any $\ep>0$,  $\Sc^\ep$ is based at infinity and $\Psi(\Sc^\ep)=\Phi(\Sc^\ep)$.  The   points  3  and  4  yield that $\Psi(\Sc)=\Phi(\Sc)$.
 \end{dem}
 
 
 \vspace{0,5 cm}
 \noindent \textbf{Acknowledgments.} It is a pleasure to thank  his PhD advisor Thierry L�vy, Franck Gabriel and Guillaume C�bron for fruitful discussions about Yang-Mills measure and the master field.  

%A \emph{rooted skein} is a finite family of loops $S=\{c_1 l_1 c_1^{-1},\ldots, c_ml_mc_m^{-1}\}$ where $\{l_1,\ldots, l_m\}$ is a skein and  $c_1,\ldots,c_m$  are paths of finite length such that 
%\begin{itemize}
%\item[i)]  $\{l_1,\ldots, l_m\}\cup \{c_1,\ldots,c_m\}$ has a finite number a transverse intersections and for pair $1\le i<j\le m,$ $c_i\cap c_j=\emptyset.$
%\item[ii)] for any $i\in \{1,\ldots,m\}$, $\underline{c_i}\in F_\infty$, $\overline{c_i} \in l_i$ and ${c_i}_{|]0,1[}\cap l_i=\emptyset$.
%\end{itemize}
%We call respectively $\{l_1,\ldots,l_m\}$ and $\{c_1,\ldots, c_m\}$ the \emph{reduction} and the \emph{branches} of $S$ and denote it by $\Rc(S)$ and $\Bc(S)$. 
%A \emph{rooted skein} $S$  is the data of  a skein  $\overline{S}$ together with  a family of paths of finite length $\Bc (S)=\{c_l,l\in \Sc\}$ such that $\overline{S}\cup \Bc(S)$ has a finite number of transverse intersections and for any loop $l\in \overline{S}$,  $\underline{c}\in F_\infty $,  $\overline{c}\in l$ and $c_{|]0,1[}\cap l =\emptyset.$
%Let us fix a rooted skein $S$ and $\Gbb$ the coarser embedded graph such that  $\Pd(\Gbb)\supset S$. We still denote by $\Vbb_s$ the set of  intersections of each loop of $\Rc(S)$ and denote by $\Vbb_f$ the set of intersection points of pair of distinct loops of $S.$ If $x\in \Vbb$

\input{BibFY}
%\nocite{*}


\end{document}
MacArxiv.tex





%\usepackage{subfig}




%	\usepackage{hyperref} 

     % Pour mettre des ent\^etes avec les titres
                              % des sections en haut de page
                              
              
%\markright{\textbf{\small{Introduction}}}                              

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\title{Comportement asymptotique de la d\'ecomposition spectrale du noyau de transition d'une marche al\'eatoire sur $\Sy_n$}

%\author{Antoine}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  

  
                              
\theoremstyle{plain}




\newtheorem{theom}{Theorem}[section] 
\newtheorem*{theomss}{Theorem} 

\newtheorem{lemma}[theom]{Lemma}
\newtheorem{prop}[theom]{Proposition}
\newtheorem{corol}[theom]{Corollary}
\newtheorem{defin}[theom]{Definition}
\newtheorem{remark}[theom]{Remark}
\newtheorem{example}[theom]{Example}




%\theoremstyle{definition}
%\newtheorem{lemm}{\lemmname}

\newenvironment{dem}{\noindent{\textit{Proof.}}}{\hfill \qedsymbol \vspace{0,2 cm}}







\newcommand{\bigslant}[2]{{\raisebox{.2em}{$#1$}\left/\raisebox{-.2em}{$#2$}\right.}}


\newcommand{\lec}{\preccurlyeq}
\newcommand{\gec}{\succcurlyeq}


\newcommand{\Inv}{\mathrm{Inv}}
\newcommand{\inv}{\mathrm{inv}}


\newcommand{\Enm}{\{-n,\ldots,-1\}\cup \{1,\ldots,m\}}
\newcommand{\di}{\mathrm{dim}}
\newcommand{\spec}{\text{Spec}}
\newcommand{\sign}{\text{sign}}
\newcommand{\iv}{\text{inv}_2}
\newcommand{\un}{\mathbf{1}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Conj}{\mathcal{C}}
\newcommand{\Bc}{\mathcal{B}}
\newcommand{\Fc}{\mathcal{F}}
\newcommand{\Kc}{\mathcal{K}}
\newcommand{\cotan}{\mathrm{cotan}}
\newcommand{\cov}{\mathrm{cov}}
\newcommand{\Var}{\mathrm{Var}}

\newcommand{\Frob}{\text{Frob}}
\newcommand{\Leb}{\text{Leb}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\E}{\mathbf{E}}
\newcommand{\V}{\mathbf{V}}
\newcommand{\af}{\mathbf{a}}

\newcommand{\vp}{\varphi}


\newcommand{\F}{\mathbf{F}}
\newcommand{\Ec}{\mathcal{E}}
\newcommand{\Lc}{\mathcal{L}}
\newcommand{\Ic}{\mathcal{I}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Hbb}{\mathbb{H}}

\newcommand{\K}{\mathbb{K}}

\newcommand{\N}{\mathbb{N}}
\newcommand{\A}{\mathbb{A}}
\newcommand{\DP}{\mathbb{H}}
\newcommand{\T}{\mathbb{T}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\Cou}{\mathrm{Cou}}
\newcommand{\id}{\mathrm{id}}
\newcommand{\Sy}{\mathfrak{S}}
\newcommand{\sk}{\mathfrak{s}}
\newcommand{\hk}{\mathfrak{h}}


\newcommand{\Gr}{\mathrm{Gr}}

\newcommand{\Y}{\mathcal{Y}}
\newcommand{\D}{\mathbb{D}}
\newcommand{\Sc}{\mathcal{S}}
\newcommand{\Sk}{\mathsf{Sk}}
\newcommand{\Ac}{\mathcal{A}}
\newcommand{\Rc}{\mathcal{R}}
\newcommand{\Gc}{\mathcal{G}}
\newcommand{\Mca}{\mathcal{M}}
\newcommand{\Cc}{\mathcal{C}}
\newcommand{\Uc}{\mathcal{U}}
\newcommand{\Vc}{\mathcal{V}}



\newcommand{\U}{\mathbb{U}}
\newcommand{\ts}{\otimes} 
\newcommand{\Bts}{\bigotimes} 

\newcommand{\Nu}{\mathcal{V}}
\newcommand{\Wg}{\text{Wg}}
\newcommand{\BnN}{{\mathcal{B}_n(N)}}
\newcommand{\Bnm}{{\mathcal{B}_{n,m}}}
\newcommand{\Bnz}{{\mathcal{B}_n(z)}}
\newcommand{\Tc}{\mathcal{T}}


\newcommand{\ep}{\epsilon}
\newcommand{\mf}{\mathbf{m}}



\newcommand{\Gb}{\mathbf{G}}
\newcommand{\Wb}{\mathbf{W}}
\newcommand{\Wgb}{\mathbf{Wg}}

\newcommand{\Gbb}{\mathbb{G}}
\newcommand{\Ebb}{\mathbb{E}}


\newcommand{\Vbb}{\mathbb{V}}
\newcommand{\Fbb}{\mathbb{F}}


\newcommand{\eb}{\mathbf{e}}
\newcommand{\vb}{\mathbf{v}}
\newcommand{\fb}{\mathbf{f}}


\newcommand{\Dd}{\mathrm{D}}
\newcommand{\Un}{\mathrm{U}}
\newcommand{\Pd}{\mathrm{P}}
\newcommand{\Adr}{\mathrm{A}}
\newcommand{\Sd}{\mathrm{S}}

\newcommand{\Vd}{\mathrm{V}}
\newcommand{\Ed}{\mathrm{E}}
\newcommand{\Ld}{\mathrm{L}}
\newcommand{\YM}{\mathrm{YM}}

\newcommand{\RL}{\mathrm{RL}}
\newcommand{\RP}{\mathrm{RP}}
\newcommand{\TL}{\mathrm{TL}}


\newcommand{\Hc}{\mathcal{H}}
\newcommand{\Ab}{\mathbb{A}}
\newcommand{\Nc}{\mathcal{N}}

\newcommand{\Pc}{\mathcal{P}}
\newcommand{\Dc}{\mathcal{D}}
\newcommand{\Xc}{\mathcal{X}}

\newcommand{\Pk}{\mathfrak{P}}
\newcommand{\Ck}{\mathfrak{C}}

\newcommand{\Oc}{\mathcal{O}}

\newcommand{\Jc}{\mathcal{J}}

\newcommand{\Wc}{\mathcal{W}}

\newcommand{\Prob}{\mathbb{P}}
\newcommand{\esp}{\mathbb{E}}
\newcommand{\SLnC}{\mathrm{SL}_n(\mathbb{C})}
%\newcommand{\SLnR}{\mathrm{SL}_n(\mathbb{R})}
%\newcommand{\SLnQ}{\mathrm{SL}_n(\mathbb{Q})}
%\newcommand{\SLnZ}{\mathrm{SL}_n(\mathbb{Z})}
%\newcommand{\SLnA}{\mathrm{SL}_n(\mathbb{A})}
\newcommand{\SLn}{\mathrm{SL}_n}
\newcommand{\SL}{\mathrm{SL}}
\newcommand{\GL}{\mathrm{GL}}
\newcommand{\SO}{\mathrm{SO}}
\newcommand{\OO}{\mathrm{O}}
\newcommand{\ON}{\mathrm{O_N}}
\newcommand{\SON}{\mathrm{SO_N(\R)}}
\newcommand{\OON}{\mathrm{O(N)}}
\newcommand{\SOON}{\mathrm{SO(N)}}
\newcommand{\SU}{\mathrm{SU}}

%\newcommand{\C}{\mathrm{C}}
\newcommand{\SP}{\mathrm{S}}
\newcommand{\Sp}{\mathrm{Sp}}
\newcommand{\SpdN}{{\mathrm{Sp}(2N)}}
\newcommand{\spdN}{{\mathfrak{sp}(2N)}}
\newcommand{\usp}{{\mathfrak{sp}(N)}}
\newcommand{\spCdN}{{\mathfrak{sp}_\C(2N)}}
\newcommand{\oN}{{\mathfrak{o}(N)}}
\newcommand{\uN}{{\mathfrak{u}(N)}}
\newcommand{\unN}{{\mathfrak{u}}}
\newcommand{\udN}{{\mathfrak{u}(2N)}}
\newcommand{\glN}{{\mathfrak{gl}_N}}
\newcommand{\GLC}{\mathrm{GL}_N(\mathbb{C})}
\newcommand{\GLR}{\mathrm{GL}_N(\mathbb{R})}
%\newcommand{\GLnQ}{\mathrm{GL}_n(\mathbb{Q})}
%\newcommand{\GLnZ}{\mathrm{GL}_n(\mathbb{Z})}
%\newcommand{\GLnA}{\mathrm{GL}_n(\mathbb{A})}
%\newcommand{\GLn}{\mathrm{GL}_n}
%\newcommand{\GL}{\mathrm{GL}}
%\newcommand{\PGLnC}{\mathrm{PGL}_n(\mathbb{C})}
%\newcommand{\PGLnR}{\mathrm{PGL}_n(\mathbb{R})}
%\newcommand{\PGLnQ}{\mathrm{PGL}_n(\mathbb{Q})}
%\newcommand{\PGLnZ}{\mathrm{PGL}_n(\mathbb{Z})}
%\newcommand{\PGLnA}{\mathrm{PGL}_n(\mathbb{A})}
%\newcommand{\PGLn}{\mathrm{PGL}_n}
%\newcommand{\PGL}{\mathrm{PGL}}
%\newcommand{\PSLn}{\mathrm{PSL}_n}
%\newcommand{\PSL}{\mathrm{PSL}}
%\newcommand{\MnZ}{\mathrm{M}_n(\mathbb{Z})}
%\newcommand{\Mn}{\mathrm{M}_n}
%\newcommand{\M}{\mathrm{M}}
%\newcommand{\ssl}{\mathfrak{sl}}
\newcommand{\so}{\mathfrak{so}}
\newcommand{\su}{\mathfrak{su}}
%\newcommand{\begin{itemize}
\newcommand{\spn}{\mathfrak{sp}_n}
%\newcommand{\PP}{\mathcal{P}}
%\newcommand{\diag}{\mathrm{diag}}
\newcommand{\vect}{\mathrm{vect}}
%\newcommand{\ppcm}{\mathrm{ppcm}}
%\newcommand{\Ind}{\mathrm{Ind}}
\newcommand{\End}{\mathrm{End}}
\newcommand{\supp}{\mathrm{supp}}
%\newcommand{\Int}{\mathrm{Int}}
\newcommand{\Ad}{\mathrm{Ad}}
\newcommand{\ad}{\mathrm{ad}}
%\newcommand{\vol}{\mathrm{vol}}
\newcommand{\ggot}{\mathfrak{g}}
%\newcommand{\h}{\mathfrak{h}}
%\newcommand{\kk}{\mathfrak{k}}
%\newcommand{\p}{\mathfrak{p}}
%\newcommand{\aaa}{\mathfrak{a}}
%\newcommand{\bb}{\mathfrak{b}}
%\newcommand{\cc}{\mathfrak{c}}
%\newcommand{\dd}{\mathfrak{d}}
%\newcommand{\ee}{\mathfrak{e}}
%\newcommand{\ff}{\mathfrak{f}}
%\newcommand{\n}{\mathfrak{n}}
%\newcommand{\s}{\mathfrak{s}}
%\newcommand{\HH}{\mathcal{H}}
%\newcommand{\Bor}{\mathcal{B}}
%\newcommand{\Cyl}{\mathcal{C}}
%\newcommand{\cN}{\mathcal{N}}
%\newcommand{\cP}{\mathcal{P}}
%\newcommand{\cR}{\mathcal{R}}
%\newcommand{\Aut}{\mathrm{Aut}}
\newcommand{\Hom}{\mathrm{Hom}}
%\newcommand{\perm}{\operatorname{perm}}
%\newcommand{\card}{\operatorname{card}}
%\newcommand{\op}{\operatorname{op}}
\newcommand{\Vect}{\operatorname{Vect}}
\newcommand{\Isom}{\mathrm{Isom}}
%\newcommand{\Out}{\mathrm{Out}}
\newcommand{\Tr}{\mathrm{Tr}}
\newcommand{\tr}{\mathrm{tr}}
\newcommand{\Id}{\mathrm{Id}}
%\newcommand{\Ker}{\mathrm{Ker}}




\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}

\renewcommand{\Re}{\mathrm{Re}\,}
\renewcommand{\Im}{\mathrm{Im}\,}

\def\build#1_#2^#3{\mathrel{\mathop{\kern 0pt#1}\limits_{#2}^{#3}}}
%c'est ce qui permet de faire la fleche avec le n\to\infty en dessous: \build{\longarrow}_{n\to\infty}^{}
1409.7793.extracted_bib
{ASMF Michael Anshelevich and Ambar~N. Sengupta. \newblock Quantum free yang-mills on the plane. \newblock {\em Journal of Geometry and Physics, 62(2):330 -- 343, 2012.
1409.7793.extracted_bib
{BP Thomas~F. Banchoff and William~F. Pohl. \newblock A generalization of the isoperimetric inequality. \newblock {\em J. Differential Geometry, 6:175--192, 1971/72.
1409.7793.extracted_bib
{TCLF Florent Benaych-Georges. \newblock Central limit theorems for the {Brownian motion on large unitary groups. \newblock {\em Bull. Soc. Math. France, 139(4):593--610, 2011.
1409.7793.extracted_bib
{BM Olivier {Bernadi and Alejandro~H. {Morales. \newblock {Counting trees using symmetries. \newblock {\em Journal of Combinatorial Theory Serie A, 123(1):104--122, 2014.
1409.7793.extracted_bib
{BianeMBL Philippe Biane. \newblock Free {Brownian motion, free stochastic calculus and random matrices. \newblock In {\em Free probability theory ({Waterloo, {ON, 1995), volume~12 of {\em Fields Inst. Commun., pages 1--19. Amer. Math. Soc., Providence, RI, 1997.
1409.7793.extracted_bib
{BianeSegalBarg Philippe Biane. \newblock Segal-{Bargmann transform, functional calculus on matrix spaces and the theory of semi-circular and circular systems. \newblock {\em J. Funct. Anal., 144(1):232--286, 1997.
1409.7793.extracted_bib
{BorotGuionnet G.~Borot and A.~Guionnet. \newblock Asymptotic expansion of {$\beta$ matrix models in the one-cut regime. \newblock {\em Comm. Math. Phys., 317(2):447--483, 2013.
1409.7793.extracted_bib
{BIPZ E.~Br{\'ezin, C.~Itzykson, G.~Parisi, and J.~B. Zuber. \newblock Planar diagrams. \newblock {\em Comm. Math. Phys., 59(1):35--51, 1978.
1409.7793.extracted_bib
{CK G.~{C{\'ebron and T.~{Kemp. \newblock {Fluctuations of Brownian Motions on GL\_N. \newblock {\em ArXiv e-prints, September 2014.
1409.7793.extracted_bib
{Collins Beno{\^{\it Collins. \newblock Moments and cumulants of polynomial random variables on unitary groups, the {Itzykson-{Zuber integral, and free probability. \newblock {\em Int. Math. Res. Not., (17):953--982, 2003.
1409.7793.extracted_bib
{CollinsGuionnetSegala Beno{\^{\it Collins, Alice Guionnet, and Edouard Maurel-Segala. \newblock Asymptotics of unitary and orthogonal matrix integrals. \newblock {\em Adv. Math., 222(1):172--215, 2009.
1409.7793.extracted_bib
{HOPS Beno{\^{\it Collins, James~A. Mingo, Piotr {\'Sniady, and Roland Speicher. \newblock Second order freeness and fluctuations of random matrices. {III. {Higher order freeness and free cumulants. \newblock {\em Doc. Math., 12:1--70 (electronic), 2007.
1409.7793.extracted_bib
{IntBM A.~{Dahlqvist. \newblock {Integration formula for Brownian motion on classical compact Lie groups. \newblock {\em ArXiv e-prints, December 2012.
1409.7793.extracted_bib
{DiaconisEvans Persi Diaconis and Steven~N. Evans. \newblock Linear functionals of eigenvalues of random matrices. \newblock {\em Trans. Amer. Math. Soc., 353(7):2615--2633, 2001.
1409.7793.extracted_bib
{Diashah Persi Diaconis and Mehrdad Shahshahani. \newblock On the eigenvalues of random matrices. \newblock {\em Jour. Appl. Prob., 31(3):49--62, 1994.
1409.7793.extracted_bib
{Driver Bruce~K. {Driver. \newblock {YM$\sb 2$: Continuum expectations, lattice convergence, and lassos. \newblock {\em {Commun. Math. Phys., 123(4):575--616, 1989.
1409.7793.extracted_bib
{AFM V.~{F{\'eray, P.-L. {M{\'eliot, and A.~{Nikeghbali. \newblock {Mod-phi convergence and precise deviations. \newblock {\em ArXiv e-prints, April 2013.
1409.7793.extracted_bib
{Franck Franck Gabriel. \newblock Planar markovian holonomy fields : construction, characterization and application. \newblock {\em Paper in preparation.
1409.7793.extracted_bib
{GrossMat David~J. Gross and Andrei Matytsin. \newblock Some properties of large-{$N$ two-dimensional {Yang-{Mills theory. \newblock {\em Nuclear Phys. B, 437(3):541--584, 1995.
1409.7793.extracted_bib
{NovakGuionnet A.~{Guionnet and J.~{Novak. \newblock {Asymptotics of unitary multimatrix models: The Schwinger-Dyson lattice and topological recursion. \newblock {\em ArXiv e-prints, January 2014.
1409.7793.extracted_bib
{HC Harish-Chandra. \newblock Differential operators on a semisimple {Lie algebra. \newblock {\em Amer. J. Math., 79:87--120, 1957.
1409.7793.extracted_bib
{HUMPHRIES Stephen~P. Humphries. \newblock On weakly distinguished bases and free generating sets of free groups. \newblock {\em The Quarterly Journal of Mathematics, 36(2):215--219, 1985.
1409.7793.extracted_bib
{IkedaWatanabe Nobuyuki Ikeda and Shinzo Watanabe. \newblock {\em Stochastic differential equations and diffusion processes, volume~24 of {\em North-Holland Mathematical Library. \newblock North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981.
1409.7793.extracted_bib
{KazakovMF V.A. Kazakov. \newblock Wilson loop average for an arbitrary contour in two-dimensional u(n) gauge theory. \newblock {\em Nuclear Physics B, 179(2):283 -- 292, 1981.
1409.7793.extracted_bib
{KazakovKostov V.A. Kazakov and I.K. Kostov. \newblock Computation of the wilson loop functional in two-dimensional u$(\infty)$ lattice gauge theory. \newblock {\em Physics Letters B, 105(6):453 -- 456, 1981.
1409.7793.extracted_bib
{ThierrySW Thierry L{\'evy. \newblock Schur-{Weyl duality and the heat kernel measure on the unitary group. \newblock {\em Adv. Math., 218(2):537--575, 2008.
1409.7793.extracted_bib
{MF Thierry {L{\'evy. \newblock {The master field on the plane. \newblock {\em ArXiv e-prints, December 2011.
1409.7793.extracted_bib
{ThierryMylene Thierry L{\'evy and Myl{\`ene Ma{\"{\ida. \newblock Central limit theorem for the heat kernel measure on the unitary group. \newblock {\em J. Funct. Anal., 259(12):3163--3204, 2010.
1409.7793.extracted_bib
{Champsmarkoholo Thierry L�vy. \newblock {\em Two-dimensional Markovian holonomy fields. \newblock Ast�risque 329, 2010.
1409.7793.extracted_bib
{MM Yu.M. Makeenko and A.A. Migdal. \newblock Exact equation for the loop average in multicolor qcd. \newblock {\em Physics Letters B, 88:135 -- 137, 1979.
1409.7793.extracted_bib
{PLM P.-L. {M{\'eliot. \newblock {The cut-off phenomenon for Brownian motions on symmetric spaces of compact type. \newblock {\em ArXiv e-prints, October 2012.
1409.7793.extracted_bib
{SenguptaYMCompact Ambar Sengupta. \newblock Gauge theory on compact surfaces. \newblock {\em Mem. Amer. Math. Soc., 126(600):viii+85, 1997.
1409.7793.extracted_bib
{IS Isadore~Manuel Singer. \newblock In Simon Gindikin, James Lepowsky, and Robert~L. Wilson, editors, {\em Functional analysis on the eve of the 21st century. {Vol. {I, volume 131 of {\em Progress in Mathematics, page 263. Birkh\"auser Boston Inc., Boston, MA, 1995. \newblock In honor of the eightieth birthday of I. M. Gel{\cprimefand, Papers from the conference held at Rutgers University, New Brunswick, New Jersey, October 24--27, 1993.
1409.7793.extracted_bib
{Hooft G.~{'t Hooft. \newblock {A planar diagram theory for strong interactions. \newblock {\em Nuclear Physics B, 72:461--473, April 1974.
1409.7793.extracted_bib
{Xu Feng Xu. \newblock A random matrix model from two-dimensional {Yang-{Mills theory. \newblock {\em Comm. Math. Phys., 190(2):287--307, 1997.
1409.7793.extracted_bib
{ZZ P.~Zinn-Justin and J.-B. Zuber. \newblock On some integrals over the {${\rm U(N)$ unitary group and their large {$N$ limit. \newblock {\em J. Phys. A, 36(12):3173--3193, 2003. \newblock Random matrix theory.