Free energies and fluctuations for the Unitary Brownian Motion

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.


Introduction
The following paper aims at studying traces of non-commutative polynomials in independant Brownian motions on the group of unitary matrices U(N ), as the size N goes to infinity. In [5,35,26,27], 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-φ convergence ( [17]). As a corollary, we obtain the fluctuations around their limit. In [28], 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. Cébron and T. Kemp obtained in [9] a similar result of gaussian fluctuation for diffusions on GL N (C), among which the U(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 [27].
Free energies of matricial models: In many random matricial models, asymptotics of E[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 ( [21,36]) 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 [20,11,7]. In the pioneering work [8], formal expansions have been proposed for several physical models. In [10,12], 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 , . . . , U q,tq ) t∈R q + be q independant Brownian motions invariant by adjunction in U(N ) (see section 2 for a definition) and denotes by Tr the usual non-normalized trace of matrices. Theorem 1.1. For t ∈ R q + and any non-commutative polynomial V in 2q variables, there exists r V > 0 and analytic functions ϕ t,V , (ψ t,V,N ) N ≥1 and ψ t,V on D r V = {z ∈ C : |z| < r V }, such that as N → ∞, where the convergence is uniform on compact subset of D r V .
For any non-commutative polynomial V in 2q variables, whose evaluation is hermitian one unitary matrices, we shall define for any integer N ≥ 1, a probability measure µ N,V on U(N ) q absolutly continuous with respect to the law of (U i,t i ) 1≤i≤q , with density proportional to e zN Tr(V (U i,t i ,U * i,t i ,i=1..q)) . Then, for any N ≥ 1, (U V N,1 , . . . , U V N,q ) denotes a random variable with law µ N,V . Theorem 1.2. If V, W ∈ C X i , Y i i=1..q are non-commutative polynomials with small enough coefficients and V * = V , then, under the probability measure µ N,V , the random variable 1 N Tr(W (U V i , U V i * , i = 1..q)) converges in probability towards a constant Φ t,V (W ).
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 [16,32,1,29,27]. Therein, we give a recursive way to compute coefficients of ϕ 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 [30] and were first proved rigorously in [27]. The Yang-Mills measure encompasses the different models for all q ∈ N * and t ∈ R q + , into one random object, for which the recursive equations has a simple interpretation. We shall use the approach of [29,27,18] by considering for any N ≥ 1, a process (H l ) l indexed by the set L(R 2 ) of rectifiable loops in the plane, valued in U(N ), whose law will be denoted by Y M N .
Planar master field: The works [1,27] proved that under Y M N , the random field ( 1 N Tr(H l )) l∈L(R 2 ) converges in probability towards a deterministic field (Φ(l)) l∈L(R 2 ) . The statement of this result first appeared in the physics literature, in the study of QCD, with the works [24,25,30], and in the mathematical paper [33], as a conjecture. The limiting field was named therein master field, following the terminology of [34]. This object is the first coefficient of the extension of the functions ϕ t,0 in Theorem 1.1. The asymptotic of 2D-Yang-Mills measure on other compact surfaces has also been investigated in the physics literature [19]. It won't be discussed in this text but could lead to future works.
Fluctuations: The study of fluctuation of traces of random elements of a compact group of large dimension started with [15], where is was investigated, thanks to representation theory tools, for the Haar measure on the classical compact Lie groups. The Theorem 1.1 allows us in particular to characterize the fluctuations in the convergence of the non-commutive distribution of a U(N )-Brownian motion towards the free unitary Brownian motion distribution. We further prove that under Y M N the random field (Tr(H l ) − E[H l ]) l∈L(R 2 ) converges in law towards a gaussian field (φ l ) l∈L(R 2 ) , characterized by the Makeenko-Migdal equations. Besides, we observe that when the loops are dilated by a factor λ, the above fields have the same gaussian behavior as λ → 0. Our result extends the work of [28], which study the gaussian fluctuations in the convergence of the empirical measure of a U(N )-Brownian motion marginal. The gaussian process obtained therein can be shown to be a deformation the one obtained in [15]. 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 [14,3], 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.
Organisation of the paper: The next section is devoted to the description of the convention we use for the standard Brownian motion on U(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. Let us define (U t ) t≥0 as the M N (C)-valued solution of the following stochastic differential equation: Lemma 2.1. i) Almost surely, for all t ≥ 0, U t ∈ U(N ).
ii) For all T ≥ 0, (U * T U T +t ) t≥0 is independant of the sigma field σ(U s , s ≤ T ) and has the same law as (U t ) t≥0 .
iii) For any t ≥ 0 and every fixed U ∈ U(N ), U U t U −1 has the same law as U t .
Proof. Let us prove the first point, the two others are left to the Reader. The processes (i √ N (K t ) p,p ) t≥0 for 1 ≤ p ≤ N and ( √ N (K t ) i,j ) t≥0 for 1 ≤ i < j ≤ 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 E[|(K 1 ) 1,2 | 2 ] = 1. Let us denote by · the symbol of quadratic variations, so that Itô's formula then yields We call this process the U(N )-Brownian motion 1 (see [13,27] for a similar definition on other classical compact groups). For N = 1, it has the same law as (e iBt ) t≥0 , where (B t ) t≥0 is the standard real Brownian motion. Let us make remarks on the scaling. Let us recall that the scalar product ·, · on u(N ) induces a Riemannian metric d on U(N ). On the one hand, this choice of metric yields that the diameter of U(N ) is d(Id, −Id) = 1 0 γ t dt, where γ : t ∈ [0, 1] → exp(tiπId N ), that is, iπId = N π. On the other hand, the law of large numbers implies that dim(u(N )) −1 K t 2 = N −2 K t 2 converges, as N → ∞, towards t. Heuristically, we may infer that, as N → ∞, for any t > 0, d(Id, U t ) behaves like K t and d(Ut,Id) d(Id,−Id) → C t ∈ (0, ∞). With this scaling, the Brownian motion "has the time to visit" 2 U(N ). Besides, the stochastic differential equation (*) does not depend 1 It can be shown that it is a diffusion on the Riemannian manifold U(N ) endowed by the left-invariant metric associated to ·, · and that its generator is the Laplace-Beltrami operator (see [23] or [28], Proposition 2.1. for an elementary proof).
2 Note that a good scaling to study the convergence of the distance in total variation d T V between the law of Brownian motion and the Haar measure, is faster than ours. Let U be a Haar distributed random variable on U(N ). It has been shown in [31] that the function t → d T V (U t log(N ) , U ) admits a cut-off around the value t = 2.
on N and such an equation makes sense in the context of free stochastic differential equations (see [5]). Let us add a last comment on the time-scale. With the above choice, the U(1)-Brownian motion appears with the same scaling in all U(N )-Brownian motions.
Lemma 2.2. For any N ∈ N * , let (U t,N ) t≥0 be a U(N )-Brownian motion. Then, the process (det(U t,N )) t≥0 has the same distribution as (U t,1 ) t≥0 .
Proof. Observe that for any N ∈ N * , (iTr(K t )) t≥0 has the same law as a standard Brownian motion. If D 2 (det) M : M N (C) 2 → C denotes the second derivative of the determinant at a point M ∈ M N (C), Itô's formula yields that What is more, Hence, and det(U 0 ) = 1.

2.2.
Free unitary Brownian motion. Let us recall the first result obtained about the behavior of unitary Brownian motion in large dimension. We shall denote by (µ N t ) t≥0 the family of random measures given by the empirical measure of eigenvalues of U t : if λ 1 , . . . , λ N ∈ U are the eigenvalues of U t , µ N t = 1 N (δ λ 1 + · · · + δ λ N ). For any polynomial function P ∈ C[X], note that tr(P (U t )) = U P (z)µ t (dz). The following theorem has first been proved in [5] using harmonic analysis on the unitary group. Theorem 2.3 ([5,35,26]). The sequence or random measures (µ N t ) N ≥0 converges weakly in probability 3 , towards a deterministic measure µ t on U, whose moments are given as follows: Using the property of independance and stationarity of multiplicative increments a U(N )-Brownian motion, together with the invariance of its law by adjunction and free probability arguments, the following Theorem was then deduced. Theorem 2.4 ([5]). For any t 1 , t 2 , . . . , t q ≥ 0 and V any non-commutative polynomial of 2q-variables, the random variables 1 N Tr(V (U 1 , U * 1 , . . . , U q , U * q )) converge in probability towards a constant.
The limiting object is called the non-commutative distribution of the free unitary brownian motion and can be characterized by the family of measures (µ t ) t≥0 together with the asymptotic freeness of the increments. This last theorem was proved in another way in [35,26,27] showing directly the convergence for any non-commutative polynomial. Let us recall how the argument of [26,27] goes to show 2.3. Let us denote by S n the group of permutations of {1, . . . , n}. For any permutation σ ∈ S n composed of #σ cycles, we define a function f σ on U(N ) by setting for any U ∈ U(N ), f σ (U ) = N −#σ Tr σU ⊗n and a function on S n , by setting for any t > 0, . Then, the latter family of functions on the symmetric group is shown to satisfy the following differential system (see [26] or Lemma 2.6).

Lemma 2.5 ([26]). For any permutation
The unique solution of this system of ordinary differential equations is a power series in 1 N and converges, as N → ∞, to a function ϕ t . It can further be shown to satisfy for any σ ∈ S n with a k cycles of length k, Setting for all t ≥ 0, n ≥ 1, µ t,n = ϕ t ((1 · · · n)), the limit in N of the former equations takes the following form: with initial condition µ 0,n = 1. This system of equations is then shown to have as unique solution given by the expression of Theorem [5]. It follows that for n ∈ N, t ≥ 0, To conclude and obtain a convergence in probability, one ultimately needs to estimate the covariances of the complex variables 1 N Tr(U n t ) n∈N,t≥0 with their complex conjugate. This latter point together with the Lemma 2.5 can be proved using the following lemma, that allows to study any polynomial in the entries and their conjugate of a unitary Brownian motion.
For any integer n ∈ N * , let us recall the left action of S n on C ⊗n , such that for any permutation σ ∈ S n and any elementary tensor v The endomorphism of (C N ) ⊗n associated to a permutation σ will be abusively denoted below by the same symbol. For any pair of distinct integers i, j ∈ {1, . . . , n}, we denote by i j the endomorphism of (C N ) ⊗n which acts like the endomorphism 1≤r,s≤N E r,s ⊗ E r,s on the ith and jth tensors and trivially on the others. Lemma 2.6 ( [13,27]). Let U t be a Brownian motion on U(N ). For any positive integers a, b, which add up to n, the following differential equation holds: For any permutation σ ∈ S n , ϕ t (σ) = N −#σ Tr(σE[U ⊗n t ]) and the Lemma 2.5 reduces to this more general one. Proof. We shall use the stochastic differential equation (*) and apply Itô formula. First, writing the u(N )-valued Brownian motion (K t ) t≥0 as a sum of independent real standard Brownian motions yields that We can now use the Itô formula to get that the variational-bounded part of the variation of the semi-martingales U ⊗2 The same analysis yields that the variational-bounded part of the variation of the semi-martingale U ⊗a

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≥0 is a u(N ) Brownian motion as defined above, for any where B t is the marginal of a standard real Brownian motion. These two naive examples suggest that the scaling chosen in Theorem 1.1 is the good one. We shall prove it in the following by estimating cumulants.
3.1.1. Scaling of cumulants. For any bounded random variable X, the function log E[e zX ] is analytic on a neighborhood of 0. We denote its analytic expansion the coefficients (C n (X)) n≥1 are called the cumulants of the random variable X.
We are interested here in the behavior in N of where the A N are bounded random matrices of M N (C), uniformly bounded in norm.
3.1.2. Cumulants. These coefficients are related to the moments of X via a Möbius inversion formula. For any n ∈ N * , the set P n of partitions of {1, . . . , n} is endowed with a partial order , such that for π, ν ∈ P n , π ν if the blocks of π are included in the one of ν. It has a maximum and a minimum that we denote respectively by 1 n and 0 n . Each partition π has #π blocks. For any sequence of complex numbers (α A ) A⊂{1,...,n} , let us set for any partition π ∈ P n , Then, there exists a unique sequence (β π,ν (α)) π ν such that for any two partitions π ν, α ν = π π ν β π,π (α).
For any π, ν ∈ P n with ν π and π = ν, we set β π,ν (α) = 0. If X 1 . . . , X n are bounded complex random variables and for any A ⊂ {1, . . . , n}, α A = E[ i∈A X i ], let us set C n (X 1 , . . . , X n ) = β 0n,1n (α) and similarly, for any pair of partitions π, ν, C π,ν (X 1 , . . . , X n ) = β π,ν (α). Then, the following expansion holds for any z ∈ C n in a neighborhood of 0, If Y and Z are bounded random variables coupled with X and if z ∈ C is in a neighborhood of 0, then The coefficient C n (X 1 , . . . , X n ) is called a cumulant and is symmetric in the variables X 1 , . . . , X n . The coefficients (C π,ν (X 1 , . . . , X n )) π ν are called relative cumulants. For any pair π ν and A ∈ ν, let us denote by A π the set of blocks of π included in A, then 3.1.3. 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 ⊗A the vector space of multilinear map on (V * ) A and for any n ∈ N * identify V ⊗{1,...,n} with V ⊗n . Any function X : {1, . . . , n} → V defines an elementary element of V ⊗A that we denote by i∈A X i . Any partition π ∈ P n defines a multilinear map A∈π V ⊗A → V ⊗n , (α A ) A∈π → A∈π α A , such that for any X ∈ V n , For any sequence (α A ) A⊂{1,...,n} such that for any A ⊂ {1, . . . , n}, α A ∈ V ⊗A , let us set for any partition π ∈ P n , Then, there exists a unique sequence (β π,ν ) π ν such that for any two partitions π ν, α ν = π π ν β π,π (α) ∈ V ⊗n .
If X 1 . . . , X n are bounded random variables valued in V on a probability space (Ω, B, P) and for any A ⊂ {1, . . . , n}, α A = E[ i∈A X i ], note that for any pair of partitions π, ν, C π,ν (X 1 , . . . , X n ) = β π,ν (α) is n-linear as a function on the space L ∞ (Ω, B, P) ⊗ V . We then define linear functions on L ∞ (Ω, B) ⊗ V ⊗n by setting for any random variables X 1 , . . . , X n ∈ L(Ω, B, P) ⊗ V , C π,ν (X 1 ⊗ . . . ⊗ X n ) = C π,ν (X 1 , . . . X n ) and C n (X 1 ⊗ . . . ⊗ X n ) = C 0n,1n (X 1 , . . . , X n ). For example, if A and B are two bounded random vectors of V, If A 1 , . . . , A n are random matrices in M n (C N ) with bounded operator norms and π, ν ∈ P n is a pair of partitions, then If σ ∈ S n is a permutation whose orbits are included in blocks of the partition ν, then Lemma 3.1. Let (T A ) A∈Pn,#A≥2 be a family of endomorphisms such that for any A ∈ P n , T A ∈ End(V ⊗A ). i) If for any A ∈ P n , then, for any pair of partitions π ν in P n , where the second sum is over sequences A 1 , . . . , A k ∈ P n , with #A i ≥ 2 for any ii) For any µ ∈ P n , let us set for each block C ∈ µ, L C = B⊂C,#B≥2 T B A . If for any t ∈ R µ and A ∈ P n , then, for any pair of partitions π, ν ∈ P n , β π,ν (α(·)) is a differentiable map and for any t ∈ R µ and C ∈ µ, d dt C β π,ν (α(t)) =
Proof. i) For any π, ν ∈ P n , letβ π,η the right-hand-side of the formula of the statement. For any ν ∈ P ν , the expansion of tensors of exponential α ν (α) equals to the sum k≥0 where the second sum is over sequences A 1 , . . . , A k ∈ P n , with #A i ≥ 2 for any 1 ≤ i ≤ k and A 1 ∨ A 2 ∨ . . . ∨ A k ν. This expression is equal to π νβ π,ν .

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 → ∞.
3.2.1. Partitioned and partial words. For each positive integer q, W q denotes the monoid of words in the alphabet made of 2q symbols x 1 , . . . , x q , x −1 1 , . . . , x −1 q . An element w of W q writes down uniquely x 1 i 1 . . . x n in , with 1 , . . . , n ∈ {−1, 1}. We call n the length of w and denote it by (w). Its p th letter x p ip is denoted by X p (w) and for any k ∈ {1, . . . , q}, we set n ± w (k) = #{r ∈ {1, . . . , n} : X r (w) = x ± k } and n w (k) = n + w (k) + n − w (k). We call partitioned word every couple (S, π), where S is a tuple (w 1 , . . . , w m ) of words of W q and π ∈ P m . Given such a couple, we set w(S) = w 1 w 2 . . . w m and (S, π) = (w(S, π)). We denote the set of partitioned words by PW q . The symmetric group S m acts diagonally on {(S, π) : S ∈ W m q , π ∈ P m }. A partial word [S, π] is the orbit of a partitioned word (S, π) ∈ PW q , with S ∈ W q , under the diagonal action of S m . We denote the set of partial words by PW q .
1. If the i th and j th letters of w belongs to the same word w k = λX i (w)µX j (w)ν, then let us set if X i (w) = X j (w) −1 and in both cases π ∈ P m+1 the partition obtained from π by substituting l with l + 1 for l > k and adding k + 1 to any block of π including k. 2. If the i th and j th letters of w belongs to two words w p = λX i (w)µ and w q = νX j (w)χ, then let us set 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 #π is equal to #π, if p and q belongs to the same block of π and #π − 1 otherwise. For any f ∈ {1, . . . , q}, we define the sets 2,w (f ) : #π = #π}. and and for any vector t ∈ R q + , w N t = w(U 1,t 1 , . . . , U q,tq ). For any partial word [S, ν] ∈ PW q , with S = (w 1 , . . . , w m ), we shall consider for where the sums are over (i, j) belonging respectively to N − 2,w(S) (f ) and N + 2,w(S) (f ). Proof. Let us fix a tuple (w 1 , . . . , w m ) of words, set w = w 1 . . . w m = x 1 i 1 . . . x n in , with 1 , . . . , n ∈ {−1, 1} and ι : {1, . . . , n} → {1, . . . , m} the map induced by the decomposition of w into w 1 , . . . , w m . For any partition ν ∈ P m , we denote by ν 0 ∈ P n the biggest partition such that ι(ν 0 ) = ν. For any σ ∈ S n , let us denote by S σ the tuple of words of the form , with (c 1 . . . c k ) cycle of σ, ordered by the position of their first letter in w. We denote W q (w) = {S σ : σ ∈ S n } and set for any S ∈ W q (w), σ S the unique permutation such that S σ S = S.
Let us denote θ : and define for any pair of distinct integers i, j two operators T + i,j and T − i,j , acting on End((C N ) ⊗{i,j} ), by setting for any M ∈ End((C N ) ⊗{i,j} ), For any collection of words S ∈ W q (w), any pair 1 ≤ i < j ≤ n and any . It follows that if U 1 , . . . , U n are U(N )-valued random variables with U i i = U j j , then for any partition π ∈ P n such that i and j belongs to the same block of π, . We shall now apply Lemma 2.6 to the tensors w ⊗A yields that for any f ∈ {1, . . . , q}, where the first and the second sums are over pairs (i, j) ∈ A 2 belonging respectively to N + 2,w (f ) and N − 2,w (f ). Consider now a partitioned word (S, ν) with S ∈ W q (w). According to (2), : 1 ≤ f ≤ q} ∈ P n and considering the evaluation against Id (C N ) ⊗n yield the two following equalities. For any partition π ∈ P n , where the second sum is over the sequences (a l , b l ) 1≤l≤k ∈ N k 2,w and π ∨ {a For any pair (a, b) ∈ N 2,w , if p, q ∈ {1, . . . , m} are such that the a th and the b th letters of w belong respectively to w p and w q , then according to (5), The two equations (*) and (**) then implie the announced formula.

3.2.4.
Scaling and asymptotic expansion of the cumulants. Let us now introduce a scaling of the above functions that matches the one of section 3.1.1 and that yields a differential system with initial conditions independant of N . For any partial word [S, ν] ∈ PW q , with S = (w 1 , . . . , w m ), the following quantity satisfies these two conditions. Let us define two operators on C PWq , by setting for any function ϕ ∈ C PWq and (S, and and ϕ 0,N ([S, π]) = 1, if π has one block and 0 otherwise. As N → ∞, the sequence of functions ϕ t,N converges pointwise towards the unique function ϕ t , such that for any t ∈ R q + and (S, and ϕ 0 = ϕ 0,1 .
For any w ∈ W q , t ∈ R q + and N ∈ N * , we shall use the simpler notation Proof. It is a direct consequence of Lemma 3.2.
Corollary 3.4. there exists a sequence of functions (ψ t,g ) g≥1 on PW q such for any (S, π), the power series with coefficients (ψ t,g ([S, π])) g≥1 has a positive radius of convergence and for N large enough, Proof. For any fixed n ∈ N * , the operators L f , D f preserve the finite dimensional space of functions supported on {x ∈ PW q : (x) ≤ n}. The above expansion follows then easily from Proposition 3.3.
In particular, for any n 1 , . . . , n m ∈ Z * , N m−2 C m (Tr(U n 1 t ), . . . , Tr(U nm t )) admits a limit as N → ∞. Together with the property of independance, stationarity and invariance by unitary adjunction of multiplicative increments of the process (U t ) t≥0 and the notion of higher order freeness developed in [12], this fact alone implies that for any (S, π) ∈ PW q , ϕ t,N ([S, π]) admits a limit as N → ∞. Nonetheless, this result does not give easily an expansion in N .

Two estimates on the cumulants
The proof of Theorem 1.1 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 6. It is nonetheless too loose to obtain a positive radius of convergence as stated in Theorem 1.1. The second type of estimates gives a much sharper bound that allows to conclude.

4.1.
All-order bounds. For any t ∈ R q + , let us define a scalar product ·, · t on R m , by setting for any a, b ∈ R m , For any word w ∈ W q , we set A t (w) = n w , n w t and for any (S, π) ∈ PW q and N ∈ N * , we define inductively a sequence by setting ψ t,0,N ([S, π]) = ϕ t,N ([S, π]) and for any g ∈ N, ) k e m(At(w 1 )+···+At(wm)) and Proof. For any p ∈ N and M ∈ M p (C), let us set M = max i∈{1,...,p} Let us fix a word w ∈ W q and denote by B w the set of partial words [S, π] ∈ PW q with n ± w(S) = n ± w . Note that B w is stable by the operations defined in section 3.2.2, so that for any f ∈ {1, . . . , q}, the two operators L f and D f preserve the finite dimensional space F w of functions on PW q with support in B w . For any F ∈ End(F w ), let us set of B w such that (S , π ) is obtained from (S, π) by a transformation of the form T ± i,j with (i, j) ∈ N 2,w(S) . It follows that the restriction of L f and D f to F w satisfy the following inequality Let us recall that for any matrices A, B ∈ M p (C), Using iteratively this formula together with the former two equations yields that for any g ≥ 1, Besides, for all λ ∈ [0, 1] and r ∈ N, on B w . Now for any words w 1 . . . w m ∈ W q with w 1 . . . w m = w, n w = n w 1 + . . . + n wm and we simply use the bound Note that these first estimates are very loose. For example, for w = x n 1 , the lemma shows that |ϕ t (w, 0 1 )| ≤ e t 1 n 2 , when we have 4 the simple bound |ϕ t (w, 0 1 )| = | U ω n µ t 1 (dω)| ≤ 1. Furthermore, for this same word, it yields for any m ∈ N * , |N m−2 C m (Tr(U n t 1 ))| ≤ e t 1 n 2 m 2 and the exponential power series of the sequence on the right-hand-side diverges.

4.2.
Sharper bounds for the first and second orders. For any positive integer m, let us denote by C m the set of Cayley trees on m vertices. For any w 1 , . . . , w m ∈ W q , T ∈ C m , we set The second estimate shows that it is optimal for tuples of words w ∈ W q , such that n w = n w .
Proof. Let us consider q independant Brownian motions (U 1,t ) t≥0 , . . . , (U q,t ) t≥0 , on U(N ), fix a tuple of m words S 0 = (w 1 , . . . , w m ) ∈ W m q , and set w = w 1 . . . w m = ) and π 0 ∈ P 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 3.2. According our choice of scaling and (2), . 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 4.1 but Then, we show that the integrands are normalized traces of unitary matrices.
Step 1:Let us remind the definition of T a,b given in (3) and (7) and that according to (8), where the second sum is over the set Υ k formed by sequences of ordered pairs and ds the Lebesgue measure on ∆ m−1 , then, for any matrices A 1 , . . . , A m ∈ M p (C), For any partition ν ∈ P m , let us set Let Ψ m be the set of strictly increasing sequences (ν i ) m i=1 ∈ P m m , with ν 1 = 0 m and ν m = 1 m . Any sequence of Υ m−1 induces an element of Ψ m , so that, rewriting the second sum of (8) thanks to the bijection (*), and then applying (11), yields that e Step 2: Now, for any partition ν ∈ P m , according to the very first equation on tensors (2.5) that we obtained, for any v ∈ End((C N ) ⊗n ) and s ≥ 0, For any partition µ ∈ P m , let us denote by µ : {1, . . . , n} → µ the belonging map and introduce ((U t,b,f ) t≥0 ) b∈µ,1≤f ≤q a collection of q#µ independant U(N )-Brownian motions. Then, let us (U µ ) µ∈Pn be a collection of independant random variables such that for each µ ∈ P n , .
For any element x ∈ Ψ m , let us denote by Υ(x) the set of sequences of pairs of integers in Υ m−1 , inducing the sequence of partitions x. For any v ∈ End((C N ) ⊗n ), let us write L(v) for the endomorphism of left multiplication by v on End((C N ) ⊗n ). It now follows from the last two formulas that Notice that for every matrices U 1 , . . . , U n ∈ U(N ) and any pair ( It remains now to unfold the above notations to get (12) To conclude, note that the normalized trace in the integrand of the right-hand-side are bounded by 1. Let Γ : Ψ m → C m be the (m − 1)!-to-one map that sends every sequence ν = (∨ k l=1 {i l , j l }) 1≤k<m to the Cayley tree with edges ({i k , j k }) 1≤k<m . Then, for every ν ∈ Ψ m , γ∈Υ(ν) |t γ | = Γ(ν)(w 1 , . . . , w m ) and the first bound of the statement follows. Another consequence of (12), is that, in the Taylor expansion of ϕ ·,N ([(w 1 , . . . , w m ), 0 m ]) around 0 ∈ R q + , the terms of degree less than m−2 vanish, whereas, the sum of terms of degree m − 1 is exactly ν∈Ψm γ∈Υ(ν) tγ (m−1)! . But, for any T ∈ C m and ν ∈ Γ −1 (T ), γ∈Υ(ν) t γ =T t (w 1 , . . . , w m ). Hence, applying Proposition 4.1 yields the second estimate.

Remark 4.3. It follows from the definition that
whereas the above proof shows that for any m ≥ 2, For any t ∈ R + , let us set and for any word w ∈ W q and t ∈ R q + , Lemma 4.4. For any words w 1 , . . . , w m ∈ W q with m ≥ 2 and any N ∈ N * , . . 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, ν] ∈ PW q , also induces recursively a partition of {1, . . . , m}, with as many blocks as ν. This partition only depends on x, we denote it by η x ∈ P m and set for η ∈ P m , B η = {x ∈ B 0 : η x = η}. For any η ∈ P m , we consider For any N ∈ N * , according to Proposition 3.3 and Duhamel's formula (9), For any t ∈ R q and x, y ∈ B 0 , let us set For any t ∈ R q + and any increasing sequence ν ∈ P l m , induced by a sequence of pairs of integers (p i , q i ) 1≤i≤l , let us set Let us fix η ∈ P m . For any y ∈ B η , a slight modification of the proof of Proposition 4.2 yields that for any t ∈ R q + , For any π ∈ P m and any linear operator A on C B 0 , let us define another operator A π by setting for any ϕ ∈ C B 0 and x ∈ B 0 , A π (ϕ)(x) = y∈Bπ ϕ(y)A(δ y )(x). The same argument yields that To conclude, we shall now expand the exponential in the right-hand-side and use triangular inequality. For each f ∈ {1, . . . , q}, let us define an operatorL f on C B 0 , by setting for all ϕ ∈ C B 0 and x = (S, π) ∈ B 0 , For any x ∈ B 0 , ν ∈ Ψ η and s ∈ ∆ #η−1 , For any tuple S of words in W q , let us denote, for each f ∈ {1, . . . , q}, by w f (S) ∈ W q , the word obtained from w(S) by deleting the letters x f and x −1 f , for f = f. Recall that for any x = [S, π] ∈ PW q , ϕ 0 (x) = 1, if #π = 1, and 0, otherwise. Then, for any Let us denote by Gathering (♦) with the last four inequalities yields that Then, the following lemma implies the annouced bound.
Proof. For any n ∈ N * , let CW 1 be the set of finite tuple S of words in W 1 , with (w(S)) = n. For any S ∈ CW 1 (n), (a, b) ∈ N ± 2,w(S) , let us define T ∓ a,b ((w)) as in 1. of section 3.2.2. For any f ∈ {1, . . . , q}, we define a linear operatorL on C CWq , by setting for any ϕ ∈ C CWq and S ∈ CW q , For any n ∈ N * , let us denote here by ϕ 0 ∈ C CW 1 the constant function equal to 1 and set for any s ∈ R + , ρ s (n) = e sL (ϕ 0 )((x n 1 )). Notice that for any word w ∈ W 1 , with (w) = n, e sL (ϕ 0 )((w)) = ρ s (n). Therefore, for any t ∈ R q According to the definition of the operatorL, the family of functions (ρ · (n)) n≥0 satifies the following differential system: for any n ∈ N * and s ≥ 0, Then, for any s ≥ 0, According to the Lemma 13 of [6], (ρ s (n)) n≥0 is the sequence of moments of a hermitian operator (therein, denoted by Λ s Λ * s ) acting on a separable Hilbert space and, according to Proposition 11 of the same article, with spectrum It follows that for all n ∈ N * , ρ s (n) ≤ λ + s n and the result then follows from (*).

5.1.
Asymptotics of the free energies. For any function V ∈ C Wq , let us set V 1 = w∈Wq |V (w)| and V ∞ = sup w∈Wq |V (w)|. We define F 1,q = {V ∈ C Wq : V 1 < ∞} and F 0,q the set of functions V ∈ C Wq , with #{w ∈ W q : V (w) = 0} < ∞. For any N ∈ N * , U 1 , . . . , U q ∈ U(N ) and V ∈ F 1,q the following sum converges almost surely and defines a random matrix Theorem. 1.1 For t ∈ R q + and V ∈ F 0,q , there exists r V > 0 and analytic as N → ∞, where the convergence is uniform on compact subset of D r V .
Proof of Theorem 1.1. For any function V ∈ F 1,q , let us define the matrix V N,t = V (U 1,t 1 , . . . , U q,tq ) and I t,V,N (z) = N −2 log E[e N zTr(V t,N ) ]. The latter analytic function satisfies on a neighborhood of 0, According to Proposition 3.3, the summand of the two sums converge pointwise. The summand of the first sum is bounded by |zV (w)|, so that this sum converges absolutly towards w∈Wq V (w)ϕ t (w)z. Each coefficient of the second power series is bounded by It follows that I t,V,N (z) = m≥1,w 1 ,...,wm∈Wq { n a , n b t }e V 1 and converges uniformly as N → ∞ on its compact subset towards a limit that we denote by ϕ t,V (z). Let us set for any V ∈ C Wq , For any m ≥ 1 and V with η V < ∞, according to Lemma 4.4, the sum is well defined and satifies . What is more, if ψ t,V,N (z) denotes the power series with coefficients (ψ N,m (V )) m≥1 , then ψ t,V,N is well defined on D r V and converges uniformly on its compact subset towards a function ψ t,V , with 1 To conclude, note that r V > r V , so that if |z| < r V < ∞, ϕ t,V (z) is well defined and the analytic function converges uniformly towards e ψ t,V (z) on D r V . denotes the tuple composed with m copies of x 1 , and (U t ) t≥0 a U(N )-Brownian motion, then For any function V ∈ C Wq and any word w = For any symmetric function V ∈ F 1,q and t ∈ R * + , the random matrix V t,N = V (U 1,t 1 , . . . , U q,tq ) is hermitian and its operator norm is bounded by V 1 . In particular, it satisfies 0 < E[e N Tr(V t,N ) ] < ∞. Let µ t,V be the probability measure on U(N ) q , whose density with respect to the law of (U 1,t 1 , . . . , U q,tq ) is E[e N Tr(V t,N ) ] −1 e N Tr(V (U 1 ,...,Uq)) . We shall denote by (U V 1,t 1 , . . . , U V q,tq ) a random variable distributed as µ t,V on U(N ) q . For any V, W ∈ C Wq , let us define for any t ∈ R q + , Then, under the measure µ N,V , the family of random variables 1 N Tr(W (U V 1,t 1 , . . . , U V q,tq )), with W ∈ F t,V converges jointly in probability towards constants Φ t,V (W ), W ∈ F t,V .
According to Proposition 4.2, dominated convergence implies that these two sequences have a limit as N → ∞.

Planar Yang-Mills measure
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.
6.1. Paths of finite length. Let us first specify the family of loops we are considering. Let us call parametrized path any Lipschitz function from [0, 1] to R 2 , that are either constant or with speed bounded by below. We denote by P(R 2 ) the set of parametrized paths up to bi-Lipshitz increasing reparametrization and call its elements paths. For any path γ ∈ P(R 2 ) with parametrization c : [0, 1] → R 2 , let us denote its endpoints c(0) and c(1) by γ and γ, and by γ −1 the reverse path parametrized by t ∈ [0, 1] → c(1 − t). For any x ∈ R 2 , we denote by L x (R 2 ) the set of paths γ ∈ P(R 2 ) such that γ = x = γ and call elements of L x (R 2 ) loops based at x. We set L(R 2 ) = ∪ x∈R 2 L x (R 2 ). For any loop l based at some point x ∈ R 2 and parametrized by the Lipschitz-continuous mapl : [0, 1] → R 2 , we call non-based loop the induced map U → R 2 up to bi-Lischitz order preserving one-to-one mapping of U. If a and b are two paths such that a = b, we denote by ab the path of P(R 2 ) obtained by concatenation.

Embedded graphs.
We call here embedded graph in the plane the data of a triple of finite sets G = (V, E, F b ), where F b are simply connected domains of the plane with disjoint interior, which boundary is the image of a non-based loops, E is a set of paths of P(R 2 ) such that the union of their image is the union of boundaries of elements of F b , V is the set of endpoints of E and the graph induced by E on V is connected. With this convention, any edge e ∈ E is either a simple loop or an injective path of finite length. We denote by F ∞,G (or simply F ∞ ) the interior of R 2 \ (∪ F ∈F bF ). We set F = F b ∪ {F ∞ } and denote by |F | the area of any element of F ∈ F b and by ∂F the non-based loop whose image is the boundary of F oriented counterclockwisely. We shall write P(G) for the set of paths that are concatenation of elements of E and L(G) (and resp. for any v ∈ V, L v (G)) for the set of loops (resp. loops based at v) in P(G). 6.3. A free group: reduced loops of an embedded graph. Let us fix an embedded graph G. For any pair of paths γ 1 and γ 2 of P(G), let us write γ 1 ∼ γ 2 and say that γ 1 and γ 2 are equivalent if one can get γ 1 from γ 2 or vice-versa by adding or erasing paths of the form e.e −1 , with e ∈ E. For any path γ, there is a unique element of minimal length in its equivalence class, that the call the reduction of γ. The set of reduced paths endowed with the operation of concatenation and reduction forms a groupoid that we denote by RP(G). For any v ∈ V, we denote by RL v (G) the set of reduced paths that are loops based at v. Endowed with the above multiplication, RL v (G) is a free group of rank #F b (we shall highlight specific free basis in section 6.6). We denote the space of multiplicative functions by M(L(R 2 ), U(N )) and by C the smallest σ-fields such that for any l ∈ L(R 2 ), h ∈ M(L(R 2 ), U(N )) −→ h(l) ∈ U (N ) is measurable, where U (N ) is endowed with its Borel σ-fields. We define as well, for any v ∈ V, the set M(L v (G), U (N )) of multiplicative functions on L v (G). For any choice of of basis Λ of RL v (G), there is a bijection It is easy to see that the preimage σ-field of the Borel σ-field by this map does not depend on this choice and we denote it by C G .

6.3.2.
Lassos basis and discrete Yang-Mills measure. For any loop v ∈ V and l ∈ L v (G), we say that l is a lasso based at v, if l = a∂ a F a −1 where a ∈ P(R 2 ), with a = v, a is a vertex in the image of ∂F and ∂ a F is the rooting of ∂F at a. We shall see in the next section that there exists basis of RL v (G) formed with lassos. Let Λ = (λ F ) F ∈F b be a lassos free basis of RL v (G), and let Y M Λ G be the law of

Lemma 6.1 ([27, 18]). i) For any lassos free basis
ii) If G is another embedded graph with P(G ) ⊂ P(G) and R G G : M(L v (G)) → M(L v (G )) denotes the restriction map, then We denote by (H l ) l∈Lv(G) the canonical process on M(L v (G), U(N )) with law YM G . The first point follows from the invariance of the law of the U(N )-Brownian motion by adjunction and the following result.
Theorem ( [22]). Let X = (x 1 , . . . , x n ) and Y = (y 1 , . . . , y n ) be two free basis for the free group F n , such that x i is conjugated to y i for all i ∈ {1, . . . , n}. Then X can be obtained from Y by a sequence of transformations of the kind (u 1 , . . . , u n ) → (u 1 , . . . , u n ) where, for some i, j, u i = u j u i u −1 j or u −1 j u i u j and u k = u k for k = i. The second point of Lemma 6.1 requires a proof (see [27,18]) that we won't reproduce here; an argument goes as follows. Let G = (V , E , F b ) be an embedded graph with v ∈ V and P(G ) ⊂ P(G). Assume that there exists F ∈ F b and F 1 , F 2 ∈ F b with F = F 1 ∪ F 2 . If λ and λ 1 , λ 2 are lassos in L v (G ) and L v (G), with faces F, F 1 , F 2 such that λ = λ 1 λ 2 , then under Y M G , H λ has the same law as U 1,|F 1 | U 2,|F 2 | , where (U 1,t ) t≥0 and (U 2,t ) t≥0 are two independant U(N ) brownian motions. Hence, it has the same law as U 1,|F | , that is the law of H λ under Y M G .
6.4. Yang-Mills measure. Let d 1 and d l be the two distances on P(R 2 ) defined in the following way: for any pair of paths γ 1 , γ 2 ∈ P(R 2 ), parametrized by c 1 , c 2 : where we have denoted by (c) the length of a path γ ∈ P(R 2 ) and the infimum is taken over all increasing bijections of [0, 1]. It have been proved in [29] that d 1 and d induce the same topology on P(R 2 ), though (P(R 2 ), d 1 ) is complete and (P(R 2 ), d ) is not. In the following, we shall only use this topology on closed space L 0 (R 2 ) and say that a sequence of paths (l n ) n≥0 converges to l if d (l n , l) → 0 and l n = l for every n ∈ N * . For any embedded graph G, with v as a vertex, let us denote by R v G : M(L(R 2 ), U(N )) → M(L v (G), U(N )) the restriction mapping. This application is measurable with respect to the σ-fields C and C G . It is shown in [29] that the measures Y M G , with G embedded graphs, can be extended in the following way.
is a sequence of paths in L(R 2 ) that converges to l, then, under Y M , H ln converges in probability towards H l . If h is an area preserving diffeomorphism of R 2 , then the process (H h(l) ) l∈L(R 2 ) and (H l ) l∈L(R 2 ) have the same law.
For any l ∈ L(R 2 ) and N ∈ N * , the random variable 1 N Tr(H l ) is called a Wilson loop.
6.5. U (1)-Yang-Mills measure. Let us consider the commutative case, N = 1. Let G be an embedded graph in the plane, v a vertex of G. For any loop l ∈ RL v (G), its winding number function define an element n l ∈ L 2 (R 2 ). Let us fix a family of lassos (λ F ) F ∈F b of G. Under U(1)-YM measure, (H λ F ) F ∈F b has the same law as #F b independent marginals of U(1)-Brownian motion (U F,|F | ) F ∈F b . Let W be a white noise on the plane with intensity given by the Lebesgue measure. The random family ( ) l∈RLv(G) and has the same law as (exp (iW (n l ))) l∈RLv(G) . For any loop l ∈ L(R 2 ), according to Banchoff-Pohl inequality (see Lemma 6.7), its winding number defines an element n l ∈ L(R 2 ). Moreover, according to Theorem 3.3.1. of [29], the map l ∈ L 0 (R 2 ) → L 2 (R 2 ) is continuous, so that, if (l n ) n≥0 is a sequence of L 0 (R 2 ) that converge for d 1 topology to a loop l ∈ L 0 (R 2 ), the sequence of random variables exp (iW (n ln )) converges to exp (iW (n l )) in distribution. Hence, the process (H l ) l∈L(R 2 ) introduced in Theorem 6.2 has the same law as (exp (iW (n l ))) l∈L(R 2 ) . Moreover, the same argument and Lemma 2.2 yield the following Lemma. Lemma 6.3. For any integer N ∈ N * , under Y M N , the law of (det(H l )) l∈L 0 (R 2 ) and (exp (iW (n l ))) l∈L 0 (R 2 ) is Y M 1 .
6.6. Two free basis of the group of reduced loops. We shall present two families of free basis of RL v (G). Let E + be an orientation of G, that is a subset of E such that for any e ∈ E, e or e −1 ∈ E + . Let us also fix a spanning tree T of the graph G and set T + the collection of positively oriented edges of T. We denote by e : F b → E + \ T + the unique bijection such that for any face F ∈ F b , e(F ) is bounding the face F . For any e ∈ E, bounding a face F , we denote by ∂ e F the loop starting with e and bounding F. For any x, y ∈ V, 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 ∈ E, It is easy to see that RL v (G) is a free group of rank #F b with free basis (β e ) e∈E + \T + . For any loop l ∈ L(G), (13) l ∼ β e 1 β e 2 · · · β en , where e 1 , . . . , e n are the edges in E \ T , used by the loop l in this order. In [27], it is proved that the second family of loops is another free basis of RL v (G). Let us give the change of basis between the two basis of RL v (G) defined above. Denote byĜ = (V,Ê) the graph dual to G: its vertices are indexed by the set of faces F, whereas its edges are indexed by the set E such that two faces F 1 and F 2 of G are neighbours inĜ, if their boundaries share a common edge. For any edge e ∈ E, we denote by F L (e) and F R (e) the edges on the left and on the right of e and denote byê the edge (F L (e), F R (e)) ∈Ê. LetT = E \ T be the dual spanning tree of T , considered as rooted at the infinite face F ∞ . We fix an orientation E + of G, such that for any edge e ∈ E + \ T , the distance inT to the root F ∞ decreases alongê. Note that for any bounded face F , F L (e(F )) = F . For any face F , we denote byT F the subtree ofT with root F and with vertices the set of childs of F inT , that we denote by C F . For any edge e ∈ E + \ T + , denote by e the order onT F L (e) induced by the time of the first visit by the clockwise contour process boarding the dual treeT starting along the left ofê −1 , as is displayed with an example in figure 3. Then, for any edge e ∈ E + \ T , denotes the product of terms increasing for e , from the left to the right. For any loop l ∈ L v (G), we denote by w l the word with letters (λ F ) F ∈F b and their inverse such that l ∼ w, given by the decomposition (13) and the inversion formula (14). Using notation of section 3.2.1, for any face F ∈ F b and any complex number z ∈ F , n w l (F ) = n l (z). 6.6.1. Complexity of lassos decompositions: We can now give an estimate on the complexity of the decomposition of a loop in G in a word of lassos associated to a spanning tree T . We display in that section results of [27] in a slightly different form adapted to our purpose. For any subset E ⊂ E and any loop l ∈ L(G), denote by L E (l) the number of times that l uses the edges of E or E −1 . Proof. The decomposition (13) and (14) yield the equality.
From now on, we shall choose the spanning tree in the following way. Lemma 6.6. there exists a spanning tree T of G such that for any face For any loop l of an embedded graph, we shall control the maximal amperean area A t (w l ) = F ∈F |F |n w l (F ) 2 with the length of the loop (l). For any loop l ∈ L(R 2 ), denote by n l the winding number function of l. The Amperean area of l is the integral Lemma 6.7 (Banchoff-Pohl inegality, [2]). For any loop of finite length l ∈ L(R 2 ), Note that ifn l = ±n l ∈ Z F , 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.
Lemma 6.8. there exists a loopl ∈ L(G), which does not use any edge twice, such that for any face F ∈ F and z ∈ F, Proof. The assumptions together with Lemma 6.5 yield that for any face F ∈ F b , Let us now choose a loopl as in Lemma 6.8. Then, A t (w l ) ≤ p 2 A(l), (l) ≤ (l 1 ) + . . . + (l m ) and Banchoff-Pohl inequality yields the expected bound. is real-valued. Let us denote by E A the set of skeins of piecewise affine loops of R 2 with transverse intersections of multiplicity at most 2. We call elements of E A affine skeins.
Proposition 6.10. For any affine skein S ∈ E A , the sequence Φ N (S) converges as N → ∞. We denote its limit by Φ(S).
Proof. For any affine skein S, there exists an embedded graph G such that the element of S belongs to L(G). Choosing an arbitrary base point v ∈ V and decomposing each loops in a lassos basis yields that under Y M N , the random family (H l ) l∈S has the same law a collection of words in marginals of independant U(N ) Brownian motions. Therefore, the Proposition 3.3 implies the result.
Proposition 6.11. Let us fix a constant K > 0. For any skein S ∈ E A of loops of length smaller than K > 0 and taking their values in a ball center of radius K, Proof. Let us assume that S = {l 1 , . . . , l m } is a family of loops in E A all based at 0. Let G S = (V S , E S , F S ) be the embedded graph with vertices the set of intersection points of the elements of S and with edges the restriction of elements of S between points of intersection. Lemma 6.9 and the second inequality of Lemma 4.1, for k = 2, imply that (15) |Φ N (S) − Φ(S)| ≤ 4π N 2 ( (l 1 ) + · · · + (l m )) 2 e 4πm( (l 1 )+···+ (lm)) 2 . Consider now (l 1 , . . . , l m ) ∈ E A satisfying the assumption of the Theorem. For any > 0, let us choose piecewise affine paths γ 1 , . . . , γ m such that γ i = l i , γ i = 0, (γ i ) ≤ K(1 + ) for any i{1, . . . , m} and {γ i l i γ −1 i , i ∈ {1, . . . , m}} is an affine skein. The application of (15) to {γ i l i γ −1 i , i ∈ {1, . . . , m}} yields the announced inequality.
This result allows then to extend the function Φ to all of Sk(R 2 ). Surprisingly, an argument analog to the proof Theorem 5.14. of [27] applies as well to the higher order case. Theorem 6.12. For any skein S ∈ Sk(R 2 ), the sequence Φ N (S) converges as N → ∞. We denote its limit by Φ(S). The function Φ is a real-valued continuous function on Sk(R 2 ). If h is an area-preserving diffeomorphism of R 2 , for any S ∈ Sk(R 2 ), Φ(h(S)) = Φ(S).
We shall also call the function Φ : Sk(R 2 ) −→ R planar master field.
Proof. For any K > 0, let Sk K (resp. E K ) be the set of affine skeins with elements included in the ball of radius 0 and with length less than K. As ∪ K>0 Sk K = Sk(R 2 ), it is enough to prove the result on Sk K . The set E K is dense in Sk K . Indeed, any loops L(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 6.2, for any N ≥ 1, the function Φ N is continuous on Sk K . Moreover, Proposition 6.11 shows that Φ N converges uniformly towards Φ on E K . Therefore, Φ N converges uniformly towards the unique continuous extensionΦ to Sk K of Φ.
It is very tempting to apply the result of section 5.1to study the Yang-Mills measure with a potential. One can obtain an improvement of Proposition 6.11 thanks to Lemma 4.4. Though, at this point, arguing as in Theorem 6.12, the present form 4.4 would only allow to study a very restricted class loops and potentials. 6.8. Small area limit . For any α > 0 and any loop l ∈ L(R 2 ), denote by α.l the image of l by the dilatation of rate α, centered at 0. If S = {l 1 , . . . , l m } is a skein, α.S = {α.l 1 , . . . , α.l m }. The following proposition shows that, as α → 0, all the quantities defined above have the same behavior as α → 0.
Proposition 6.14. The following Taylor expansions are true for any N ∈ N * . As α → 0, for any loop l ∈ L(R 2 ), and for any skein S with at least two loops, where the sum is over connected graph with vertices S and #S − 1 edges. In both cases, there exists a positive continuous function b, independant of N , such that O(α |S| ) ≤ α |S| b( l∈S (l)).
Proof. If S ∈ E A , the assertion is a direct consequence of Proposition 4.2. Continuity of the functions Φ N , Φ and b allows then to conclude.
A direct consequence is the following Corollary 6.15. Let W be a white noise on R 2 , with intensity given by the Lebesgue measure. As t → 0, the Gaussian field ( 1 t φ t.l ) l∈L(R 2 ) , as well as, for any N ∈ N * , the random family (t −1 (Tr(H t.l ) − N Φ(t.l))) l∈L(R 2 ) , under Y M N , converge in distribution towards the gaussian field (iW (n l )) l∈L(R 2 ) .

Makeenko-Migdal equations
We shall now address the problem of the computation and characterization of the master field. We say that S ∈ Sk(R 2 ) is a 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 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 Sk r (R 2 ), that we show how to compute it inductively. 7.1. Makeenko-Migdal equations for the master field on skeins. For any skein S, let us denote by W S the expectation E Y M N l∈S Tr(H l ) , we call this function a Wilson skein 5 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 S using Itô formula to yield a first order differential system for the family (W S ) S∈Skr(R 2 ) , with areas of the faces of a graph G containing S as variables. However, this differential system yields at first sight non-regular Wilson skeins W S on M(P(G), U(N )), as features the example 7.1.
Example 7.1. 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 4 in dashed lines. In this basis, the loop is decomposed as l = l C l B l 2 A l B l A . Using Itô formula as described in Lemma 2.6 and differentiating with respect to the area of the faces C and B yields d d|C| W l = − W l 2 and 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: For any regular skein S, one must therefore face the problem of finding a closed system of Wilson skeins containing W S . The system of equations given by Lemma 2.6 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 [27], 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 S be a skein and x be a point of intersection of its elements (between themself or each other). Let us denotes by S x the skein composed with the same loops as S 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 S, turns at the point x using the other strand with the same orientation (see figure 5).
The following proposition is proved in [27], in a more general framework and relies on integration by parts applied to the product of a function on M(P(G), U(N )) with the density of the discrete Yang-Mills measure. We provide here another proof relying on the decomposition in lassos described in section 6.6 and on the invariance of the Brownian motion by adjunction. Let us fix a regular skein S and an embedded graph G such that elements of S belongs to P(G).
Proposition 7.2 (Makeenko-Migdal equation). Let F 1 , . . . , F 4 be the four faces of G around a point of intersection x ∈ V of S in a cyclic order and such that F 1 is the face bounded by the two edges of S leaving x. Then, For any skein S, let us set n S = l∈S n l . Notice that for N = 1, the equality (16) is equivalent to the fact Proof. Let us denote by µ x the operator d where the faces are numbered as in the Proposition. The strategy of our proof is to choose a spanning tree of G to decompose the loops of the skein as described in section 6.6 and to use Lemma 3.2 in such a way that terms on the left-hand-side of (16) cancel themselves so that a single cut and join transformation contributes.
A differential equation indexed by skeins: Let us fix a vertex v ∈ V, a spanning tree T of G and choose the lassos basis Λ of RL v (G) associated to T and v, as in section 6.6. Decompose each loop l ∈ S as in (13) and use the inversion formula (14) 6 as is done in the example 7.1 for a single loop. Let us order arbitrarily the elements of S and denote by S = (w l 1 , . . . , w lm ) the family of words in lassos associated to element of S, set w = w l 1 . . . w lm and n = (w). Let us denote by t ∈ R F + , the vector of bounded faces area. According to the definition of the discrete Yang-Mills measure given in section 6.3.2 and recalling the notation of section 3.2.3, W S = K t (S, 1 #S ).
By assumption, each edge e ∈ E + is used at most once by a General compensations: Consider two faces A and B such that A is the first child of B inT B for e (B) . Observe that if the letter λ B (resp. λ −1 B ) occurs in the words w l 1 , . . . , w lm , then it is always on the left (resp. on the right) of λ A (resp. λ −1 A ). It follows that for any pair of distinct edges e 1 , e 2 such thatê 1 ,ê 2 ∈ [B, F ∞ ]T , the reduction of the elements of T Compensations for a specific spanning tree: Consider now four faces satisfying the assumptions of the Proposition. Thanks to the "locality" of the relation (16) 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 ∞,G and that they do not disconnectĜ, that is, the graph is connected. Let us order the faces F 1 , . . . , F 4 counterclockwisely and choose a spanning tree T of G such that F 2 and F 3 are leaves ofT and such that (F 3 , F 4 , F 1 ) and (F 2 , F 1 ) are paths ofT , as is displayed in figure 6. Observe that the orientation of the three edges having x as endpoints and belonging toT , that is induced by the loops of S, 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 e with e an edge such that F 1 , F 2 ∈T F L (e) (resp. F 3 , F 4 ∈T F L (e) ). One would like now to apply twice Figure 6. A spanning tree ofĜ 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 e , with e an edge such that F 1 , F 2 ∈T F L (e) (resp. F 3 , F 4 ∈T F L (e) ).

FREE ENERGIES AND FLUCTUATIONS FOR THE UNITARY BROWNIAN MOTION 41
formula (18) and compare Instead, let us remark the following.
For any face A, B ∈ F b , let us denote by Θ A,B the automorphism of the free group RL v (G) that maps λ B to λ A λ B λ −1 A and fixes λ F for F = B. According to Lemma 6.1, for any skein S with elements in L v (G), The automorphism Θ A,B transposes the letters λ A and λ B in the decomposition of l 1 , . . . , l m in Λ. In particular, when A and B are two faces such that A is the first child of B inT B for e(B) formula 18 also applies with Θ A,B (S) in place of S. Notice that the letters λ F 2 and λ F 3 are consecutives in the decomposition of elements Θ F 3 ,F 4 (S) in Λ (this is not true for S) . Therefore, for any edge e ∈ [F 2 , F ∞ ] used by S, (19) T The equalities (19), together with (e(F 3 )) = (e(F 2 )) = 1, yield that the equation (20) gets simplified into If x is an intersection point of the loop l 1 , then Moreover, if F ∈ F b is a neighbour face of the unbounded face, then Proof. For any skein {l 1 , . . . , l m } of m loops and any π, V ∈ P m with π ≤ V, let us and C π,V = C π,V (Tr(H l i ), i ∈ {1, . . . , m}). We shall consider the normalized cumulant Φ π (l 1 , . . . , l m ) = B∈π Φ(l i , i ∈ B) = N m−2#π C π (l 1 , . . . , l m ).
Observe that equations (*) and (***) on Φ N do not depend on N .

Generalized Kazakov basis.
For any regular skein S, denote byG S = (V S , E S , F S ) an embedded graph such that S ⊂ P(G S ) minimizing #E S and #V S . We warn the Reader that the image of edges ofG S are not necessarily contained in the image of loops of S and thatG S is not always unique. Consider G S = (S, E S ) the graph with vertices indexed by S, such that two loops are connected in G S if and only if they intersect each other. Then, images of edges ofG S are included in the images of loops of S if and only if G S is connected. In that case,G S is the unique finest embedded graph such that S ⊂ P(G S ). In this section, we shall fix a skein S such that G S is connected and set G =G S . Let E + and λ be respectively the orientation and the permutation of the edges E induced by S.
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 7.3. To that purpose, let us set .
For any loop l ∈ L(G), denote respectively by n l ∈ C F and δ l ∈ C E + the winding number function of l and the function The following lemma is proved in [27](Lemma 6.28.).
ii) The image of µ is the orthogonal space to { * v , v ∈ V} ∪ {δ l , l ∈ S}.
iii) The intersection of linear spaces spanned by { * v , v ∈ V} and {δ l , l ∈ S} is C1 E + .
We have a first answer to our question: for any regular skein S, the vector space spanned by the operators {µ(∇ a )(e), e ∈ E + } is not span({ d d|F | , F ∈ F b }). Nonetheless, for some skeins, the third condition (***) allows to complete the lacking information. is an isomorphism.
We call a skein satisfying the condition of the Lemma a skein based at infinity. Note that each vertex of G has degree 4, hence #E = 2#V and Euler relation implies #F = #V + 2. Therefore dim(Im (µ)) = #V − m + 1. For any vertex v ∈ V, let e 1 (v) and e 2 (v) be the two outgoing edges at v ordered clockwise and set Let us denote respectively by V s and V f the set of self-intersection points of each loops and the set of intersection points of pair of distinct loops of S. The type of crossing induces on V f an equivalence relation ∼ such two points x and y are equivalent if they belong to the same loops. For any pair of distinct points x, y ∈ V f such that x ∼ y, denote by β x,y the function on E + that is α x + x,y α y , where x,y = 1, if e 1 (x) and e 1 (y) belong to different loops and −1 otherwise.
Lemma 7.6. i)For any v ∈ V s and any pair of vertices x, y ∈ V f belonging to the the same pair of loops, α v , β x,y ∈ Im (µ).
ii) Let T f be a spanning acyclic directed subgraph of the complete graph on V f such that connected components of T f are the equivance class of ∼ on V f . The family {α v : v ∈ V s } ∪ {β x,y : (x, y) ∈ T f } is a free family of Im (µ) and is a basis if and only G S is a tree.
Proof. i) Indeed, for any v ∈ V s and x, y ∈ V f belonging to the same two of loops, the vectors α v and β x,y are orthogonal to { * v , v ∈ V} and {δ l , l ∈ S}.
ii) It is easy to see that thanks to the acyclycity of the graph T f , the family where m is the number of connected components of T f . The latter are in bijection with edges of G S . Recall that G S is connected so that m = m − 1 if and only if G S is a tree. To conclude, recall that dim(Im (µ)) = #V − m + 1.
If S is made of a pair of loops that intersect each other, then the conclusion of ii) of Lemma 7.6 trivially holds, we give an example of such a basis in figure  8. There is a choice of directed acyclic graph that makes the decomposition in the basis easier, namely in each equivalence class of V f , choose a base point and connect any other point towards it. If R S is a set of class representatives for ∼, for any v ∈ V f , we denote by v the unique element of R S equivalent to v. .
is a free family and for any function ϕ belonging to its span, These free-families have the following pre-image under the function µ. For any v ∈ V 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 V 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 9, where we draw the pre-image of the family described in figure 8.  If G S have cycles, we shall complete 7 the basis given by Lemma 7.6 in the following way. For each edge (l 1 , l 2 ) ∈ E S , we denote by v l 1 ,l 2 the element of R S at the intersections of l 1 and l 2 . For each edge (l, l ) ∈ E S \ T S , consider the unique path (l 1 , l 2 , . . . , l k ) in T S from l to l and for any i ∈ {1, . . . , k}, let c i be the regular path that is the restriction of l i+1 such that c i = v l i ,l i+1 , c i = v l i+1 ,l i+2 (where loops are indexed by Z/kZ). The concatenation c 1 c 2 . . . c k is a loop of G, that follows elements of the cycle (l 1 , l 2 , . . . , l k , l 1 ) and change from one strand to another at the points of R S . The application µ maps the loopl l,l to γ l,l = µ(l l,l ) = 1 α v l 1 ,l 2 + 2 α v l 2 ,l 3 + . . . + k α v l k ,l 1 , where for any i ∈ Z/kZ, i = 1, if e 1 (x i ) ∈ l i+1 and −1 if e 1 (x i ) ∈ l i (with the above notation, i = v l i−1 ,l i ,v l i ,l i+1 ). Let us write l ,l = k .
is a basis of Im (µ). Moreover for any function ϕ ∈ C E + belonging to Im (µ), Proof. First, we should notice that #R S = #E S = m − 1 + #E S \ T S and that this family has the good cardinality: To conclude, we check that the relation (K) holds true for any function ϕ in the span of this family. We denote by R the subset of R S representing classes indexed by E S \ T S . Let c ∈ C V\R S ∪R be a vector such that ϕ = v∈Vs c v α v + v∈V f \R S c v β v,v + (l,l )∈E S \T S c v l,l γ l,l . Let r belong to R and let (l, l ) be the edge E S \ T S such that r ∈ l ∩ l . Then, γ l,l (e 1 (r)) = l ,l and for any v ∈ V f \ R S , such that v = r, β v,r (e 1 (r)) = v,r , whereas e 1 (r) cancels any other element of the family. What is more, for any v ∈ V \ R S , γ l,l (e 1 (v)) = 0. This computation immediatly yields the formula (K).
Let us give the simplest example, where G S is not a tree, that is when S is made of three cycles that intersect each others twice. The graph G S is a triangle, we choose as set of representatives the three points of intersections lying on the boundary of F ∞ . We draw on figure the families of loops one get when the circles have the same orientation.
Let us see how the above construction answer our initial question. For any loop l ∈ S, µ(n l ) = l 1 F l ∈ C F ∞,1 , where l = n l (F l ) ∈ {−1, 1}. Hence, for any set of class representatives R S for ∼ and T S a spanning tree of G S , the family {n l v,v , v ∈ V S \ R S } ∪ {n lv , v ∈ V s } ∪ {nl l,l : (l, l ) ∈ E S \ T S } ∪ S ∪ {1 F } is a Figure 10. Here is the loops that are mapped to the basis β and γ, for three circles counterclockwisely oriented. The class representative of ∼ are taken on the boundary of F ∞ . In this case, the loop l l,l does not depend on the choice of the spanning tree of G S and is drawn with dashed lines, whereas the familly β has three elements.
basis of C F and its image under µ is the free family {α v , v ∈ V s } ∪ {β x,y , (x, y) ∈ T f } ∪ {γ l,l , (l, l ) ∈ E S \ T S } completed by the canonical basis of C F ∞,1 . Moreover, for any ϕ ∈ Im (µ) ⊕ C F ∞,1 ⊂ C E + ⊕ C F ∞,1 , ( ) Using Theorem 7.3, we have an expression for all area derivatives of Φ(S), for any skein based at infinity and such that G S is connected. What is more, note that if S is based at infinity and G S is not connected, then Φ N (S) = 0. Nonetheless, observe that if v ∈ V s , it may happen, as in example of figure 9 at the only point of V s , that among the two skeins S L v and S R v obtained by splitting S 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 Φ. 7.3. Complexity of skeins. We shall consider in this last section a slightly different notion of embedded graph. We call a multi-embedded graph in the plane a triplet G = (V, E, F b ) satisfying the same conditions as an embedded graph as defined at the beginning of section 6.2 but where the element of F b are allowed to be non-simply connected.
Let us fix a regular skein S. We let G S = (V S , E S , F S ) be the finest multiconnected embedded graph such that S ⊂ P(G S ). The graph G S is connected if and only if G S is connected. For any l ∈ P(G S ), we consider    and if v ∈ V s , for any partition S L v S R v = S v , max{C(S L v ), C(S R v )} < C(S). ii) For any regular skein S, there exists a family (S ) >0 of skeins based at infinity with C(S ) = C(S) for any > 0 and a family of smooth paths (r l ) l∈S such that for any l ∈ S, r l ∈ l and S converges to {r l lr −1 l , l ∈ S}, as → 0.
Proof. i) Assume that v ∈ V f . Then, for any loop l ∈ S, that does not contain v, l ∈ S v and d ∞,Sv (l) ≤ d ∞,S (l). If l 1 and l 2 are the two loops crossing at v, then d ∞,Sv (l 1 • v l 2 ) ≤ min{d ∞,S (l 1 ), d ∞,S (l 2 )}. Moreover, V s (S v ) = V s (S) and for any w ∈ V s (S), d ∞,Sv (l v ) ≤ d ∞,S (l v ). Therefore, the fact that I(S v ) = I(S) − 1 yields the expected inequality. Assume now that v ∈ V s . Let l ∈ S be the loop of S crossing at v. Recall that l v,L and l R v 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 S L v S R v of S v separating these two loops. For any loop l ∈ The right side needs more caution. Consider the paths c ± ∈ P(Ĝ Sv ) 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 ∞,G Sv . Their combinatorial length satisfies (c + ) + (c − ) ≤ 2 + I Sv (l L v ) + 2 (c). Therefore, The number of intersections of the right skein S R v is bounded by I(S \{l})+I Sv (l R v ). Moreover, for any loop l ∈ S The equality I Sv (l R v ) + I Sv (l L v ) + I(S \ {l}) = I(S v ) = I(S) − 1 concludes. ii) For each l ∈ S, such that d ∞,S (l) > 0, consider a self-avoiding path c l ∈ P(Ĝ S ) such that F ∞,G l ∩ c l = ∅ and c l = F ∞ (G S ). Choose such a family (c l ) l∈S of loops that do not cross each other but may be merged with one another. Deform each loop l along c l intol so that the deformation intersects exactly twice each dual edge of c l and does not intersect the deformation of other loops. Denote bỹ S the skein {l : l ∈ S, d ∞,S (l) > 0} ∪ {l : l ∈ S, d ∞,S (l) = 0}. By construction, for any l ∈S, d ∞,S (l) = 0 and I(S) = I(S) + 2 l∈S (|c l | − 1) = C(S).
Each path c l ∈ P(Ĝ S ) induces a pathc l ∈ P(ĜS) such that faces belonging toc l are boarded byl. When the areas (|F |) F ∈∪ l∈S c l go to zero, the vector of random variables (H l ) l∈S converges in distribution to (H r l lr −1 l ), where r l are smooth paths such that r l ∈ l. 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 S, then µ 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 S and S , let us say that S and S are equivalent and write S ∼ S if there exists a family {c l , l ∈ S} of paths of P(R 2 ) such that c l ∈ l for any l and S = {c l lc −1 l , l ∈ S}. Theorem 7.13. there exists a unique function Φ on Sk(R 2 ) satisfying the following equations. 1. Φ({1}) = 1. 2. If S − and S + are two skeins that are separated by a closed Jordan curve, Φ(S − S + ) = 0. 3. Φ is continuous for the topology of 1-variation. 4. If S ∼ S , then Φ(S ) = Φ(S). 5. For any area-preserving diffeomorphism g of the plane, Φ • g = Φ. 6. For any regular skein S, Φ is differentiable with respect to (|F |) F ∈F S and satisfy the following differential equations. If x is the intersection of two different loops, If x is the intersection of a loop of S with itself, For any face F ∈ F S , neighbour of F ∞ ,