Alice Guionnet
email: guionnet@math.mit.edu.

Free analysis and random matrices

Keywords: Random matrices, non-commutative measure, Schwinger-Dyson equation. Mathematics Subject of Classification : 15A52 (46L50, 46L54)

We describe the Schwinger-Dyson equation related with the free difference quotient. Such an equation appears in different fields such as combinatorics (via the problem of the enumeration of planar maps), operator algebra (via the definition of a natural integration by parts in free probability), in classical probability (via random matrices or particles in repulsive interaction). In these lecture notes, we shall discuss when this equation uniquely defines the system and in such a case how it leads to deep properties of the solution. This analysis can be extended to systems which approximately satisfy these equations, such as random matrices or Coulomb gas interacting particle systems.

Introduction

Dan Voiculescu gave the Takagi lectures [START_REF] Voiculescu | Aspects of free analysis[END_REF] in 2007 on some aspects of free analysis. He emphasizes the role of the free difference quotient that he introduced in [START_REF] Voiculescu | The coalgebra of the free difference quotient and free probability[END_REF]. It is the natural derivation for non-commutative variables with the highest degree of non-commutativity, as introduced in free probability theory. The free difference quotient is as well the central object of these lectures notes, as we discuss the analysis of equations, called Schwinger-Dyson (or loop) equations, that it governs. These equations appear in different domains in combinatorics, operator algebras, or probability theory. In free probability, they play the role of an integration by parts for non-commutative laws. A classical analogue of such equations is given by

x f (x)dγ(x) = f (x)dγ(x) (1)
for a large set of test functions, for instance polynomial functions. If we add the condition 1dγ(x) = 1, we have a unique solution. For instance we can compute recursively the moments and see that they are given by the number of pair partitions. In fact this equation characterizes the Gaussian law and it is often used to show convergence to this law, for instance by 1 Stein's method. We shall consider similar equations but in a non-commutative setting. They are often derived by integration by parts or, more generally, by using Gauge invariance and perturbations. As in the classical setting [START_REF] Anderson | An introduction to random matrices[END_REF], the Schwinger-Dyson equations provide a bridge between combinatorics and analysis. Integration by parts can ultimately be used to uniquely define the density of the measure in classical probability theory. In non-commutative probability theory, there is no notion of density, and integration by parts is a legitimate replacement for that. However, Schwinger-Dyson equation does not in general specify the non-commutative law uniquely. Also, unlike in classical probability theory, it is not easy to say much about the properties of a law satisfying such a Schwinger-Dyson equation. In fact, in such a generality, the only properties of such a law that are known were proved by Y. Dabrowski [START_REF] Dabrowski | A note about proving non-Γ under a finite non-microstates free Fisher information assumption[END_REF]. He considers the von Neumann algebra associated with the solution by the Gelfand-Naimark-Segal construction (this is a non-commutative analogue of the space of bounded measurable functions). He showed the non-Gamma property (there are no non-trivial asymptotically central sequences of elements) and factoriality (the associated von Neumann algebra has a trivial center). Under a further technical assumption, Y. Dabrowski and A. Ioana [START_REF] Dabrowski | Unbounded derivations, free dilations, and indecomposability results for II 1 factors[END_REF] proved that there are no Cartan subalgebras. However, one could expect much more. For instance, in the classical case, if the density of a probability measure is smooth and strictly positive, one can build it as a smooth transport of the Lebesgue measure, hence showing that the associated C * and W * algebras are isomorphic. One of the goal of these notes is to develop techniques to study Schwinger-Dyson equation and its solutions, and eventually to investigate such questions. To be more precise, let ∂ X i be the free difference quotient given, for any choice of indices j 1 , . . . , j k ∈ {1, . . . , d}, by

∂ X i X j 1 • • • X j k = j =i X j 1 • • • X j -1 ⊗ X j +1 • • • X j k .
Let µ be a linear functional on the set of polynomial functions in d non-commutative indeterminates. We say that µ satisfies the Schwinger-Dyson (or loop) equation iff

µ ⊗ µ(∂ X i P) = µ(PJ i ) (2)
for any polynomial P and some given polynomial (or more general) functions

(J i , 1 ≤ i ≤ d).
This can be understood as an integration by parts, with the notable difference from the classical case, that the difference quotient takes values in the tensor product. In the classical case where ∂ X i is the classical derivative, such an equation has a unique solution and J i = -∂ x i log dµ dx i . Knowing J i , 1 ≤ i ≤ d, defines uniquely µ and in fact gives all properties you might wish to know about µ. Even uniqueness of solutions to Schwinger-Dyson equation is not true in general in the non-commutative setting, and we shall first investigate this question. Then we shall study properties of these solutions and mainly prove that in the domain where we can prove uniqueness, we can also build transport maps between solutions. Namely assume that J i = D X i V with D X i the cyclic derivative:

D X i X j 1 • • • X j k = j =i X j +1 • • • X j k X j 1 • • • X j -1 . If V = X 2
i , it is is easy to see that there is a unique solution to the corresponding Schwinger-Dyson equation, called the law of d semi-circle laws, and denoted by σ d . If V is a small perturbation of the quadratic case or even strictly convex, we can prove uniqueness of the solutions, denoted by µ V . Then we can prove that there are smooth functions (namely absolutely converging series) T 1 , . . . , T d ant T 1 , . . . , T d such that for all polynomial

P µ V (P) = σ d (P(T 1 (X 1 . . . , X d ), . . . , T d (X 1 . . . , X d ))) σ d (P) = µ V P(T 1 (X 1 . . . , X d ), . . . , T d (X 1 . . . , X d )) .
Such a property allows to show that the associated C * and von Neumann algebras are isomorphic. It turns out that generating functions for planar maps also satisfy such Schwinger-Dyson equations. This is reminiscent to the characterization of the Gaussian law as the law whose moments count pair partitions. Ultimately, this allows to identify the moments of solutions with generating functions of planar maps, and eventually transport maps could be used to compute critical exponents of the latter. Moreover, this property has several other consequences: the associated C * -algebras are projectionless, and in particular self-adjoint polynomials have connected support [START_REF] Guionnet | Free diffusions and matrix models with strictly convex interaction[END_REF]. Furthermore, the associated von Neumann algebras and C*-algebras have the so-called Haagerup property [START_REF] Guionnet | Free diffusions and matrix models with strictly convex interaction[END_REF].

There are also domains in probability theory where the Schwinger-Dyson equation appears, but only at a large parameter (a dimension) limit. For instance, we can construct random models such that the Schwinger-Dyson equation is true, but only in average, or/and up to a small error. This is the case for matrix models which are laws on N × N random matrices where typically the linear functional given by the normalized trace 1 N Tr satisfies

E[ 1 N Tr ⊗ 1 N Tr(∂ X i P)] = E[ 1 N Tr(PD X i V )] (3)
for all polynomial P and i ∈ {1, . . . , d}. It turns out that one can prove in some cases that 1 N Tr(P) self-averages (i.e is close to its expectation) and therefore any limit point satisfies the Schwinger-Dyson equation [START_REF] Bekerman | Transport maps for β-matrix models in the multi-cut matrix regime[END_REF]. When uniqueness of the solutions to the latter holds, we deduce that it converges towards this limit. But in fact Schwinger-Dyson equation can be used even further to find a complete expansion for E[Tr(P)] in terms of N -1 . Indeed, the large dimension behavior of such systems can be analyzed thanks to [START_REF] Bekerman | Transport maps for β-matrix models and universality[END_REF], by expanding the observables around their limit, in the spirit of Stein method or, more simply, perturbation theory. These are called topological expansion as in the case of random matrices, each coefficient of the series corresponds to a generating function for the enumeration of maps of a given genus. Moreover, the ideas related to transport maps can be developed in this setting as well, yielding approximate transport maps (that is functions which map one probability to the other, up to a small error). The latter allows to derive fine properties of the eigenvalues such as the universality of the fluctuations of the eigenvalues. To be more precise let us consider the so-called β-models

given by the probability on R N :

dP V N,β (λ 1 , . . . , λ N ) = 1 Z N i< j |λ i -λ j | β e -N V (λ i ) dλ i .
The case β = 2 corresponds to the random matrix model introduced above with d = 1: Equation ( 3) can be checked with 1 N Tr replaced by the empirical measure of the λ i 's. The fluctuations of the variables (λ i ) 1≤i≤N in the case V (x) = βx 2 /2 were studied first in the case β = 2 by Tracy and Widom [START_REF] Tracy | Level-spacing distributions and the Airy kernel[END_REF][START_REF] Tracy | Level spacing distributions and the Bessel kernel[END_REF][START_REF] Deift | Random matrix theory: invariant ensembles and universality[END_REF] based on the integrable structure of the law, and much more recently for all β's by Ramirez, Rider, Virag [START_REF] Ramirez | Beta ensembles, stochastic airy spectrum, and a diffusion[END_REF]. It is shown that if we order the eigenvalues λ 1 < . . . < λ N , the largest eigenvalue fluctuates around its limit 2 like N 2/3 and the limiting law is denoted TW β . Inside the bulk, the correlation functions converge vaguely towards the Sine β distribution. This entails the convergence in average of the spacings distribution N(λ iλ i-1 ) to the Gaudin distribution when β = 2 [START_REF] Tao | The asymptotic distribution of a single eigenvalue gap of a Wigner matrix[END_REF]. We can construct approximate transport maps from P βx 2 N to P V N for a large family of nice potentials V . Controlling the large dimension behavior of this transport map, we can prove that the fluctuations of the eigenvalues are like in the case V = βx 2 for any of these nice potentials. This kind of arguments can be generalized to several matrix models.

Hence, Schwinger-Dyson equations seem to be a common theme in several problems, where it can be used to analyze fine properties such as isomorphisms and universality. Uniqueness of solutions to Schwinger-Dyson equation allow to show that apparently different quantities such as generating functions for the enumeration of planar maps and asymptotics of matrix integrals are related, as both quantities satisfy these equations. In fact, topological expansions can also be retrieved from Schwinger-Dyson equations through the so-called topological recursion which are based on the free difference quotient, see e.g. the discussion in section 4.1. These topological recursions seem to appear in many other domains, see e.g. the work of B. Eynard et al, see e.g. [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF][START_REF] Borot | All-order asymptotics of hyperbolic knot invariants from nonperturbative topological recursion of A-polynomials[END_REF].

We will first describe domains where Schwinger-Dyson equation shows up. We start by relating it with the problem of enumerating planar maps. Subsequently, it represents integration by parts in the context of non-commutative variables. This can be extended to other settings such as planar algebras and/or type III factors [section 2.8]. The fact that Schwinger-Dyson equation appears in seemingly different domains allows to connect them, moments of noncommutative variables being often described by combinatorial objects, namely planar maps. There are other domains in probability where such an equation appears, at least asymptotically. This is the case of random matrices, or particles systems in repulsive interaction. This allows to derive topological expansion (that is large dimension expansions of the observables)[section 4.1] and universality (that is criteria to show that fluctuations of some observables do not depend much on the model) [sections 4.3 and 4.3.2] .

The idea developed in these lecture notes are inspired by Voiculescu's program to develop free probability as a generalization of classical probability for non-commutative variables, en-capsulating the central notion of freeness which replaces independence. The analogy between these two domains allows to bring ideas from one to the other. In the intermediate level, random matrices are random objects which converge towards non-commutative variables: as such, understanding their limit allows to analyze them. Reciprocally, random matrices can be used to find out how to adapt concepts from classical probability to the free set-up. We here illustrate this point with the example of transport maps, which are developed in free probability and random matrix theory based on ideas going back to Monge and Ampère.

2 The free difference quotient and the Schwinger-Dyson equations

The free difference quotient

The free difference quotient ∂ X was defined by D. Voiculescu as a natural derivation in the context of non-commutative variables. Let X 1 , . . . , X d be non-commutative indeterminates. We will assume that the space of polynomials C X 1 , . . . , X d is equipped with an involution * and that the indeterminates X 1 , . . . , X d are self-adjoint with respect to this involution; X i = X * i , 1 ≤ i ≤ d. Hence, for any complex number z, any j 1 , . . . , j k ∈ {1, . . . , d}

(zX j 1 • • • X j k ) * = zX j k • • • X j 1 .
The free difference quotient with respect to the variable X i is then given, for any choice of indices j 1 , . . . , j k ∈ {1, . . . , d} by

∂ X i X j 1 • • • X j k = j =i X j 1 • • • X j -1 ⊗ X j +1 • • • X j k .
In other words, ∂ X i satisfies the Leibniz rule

∂ X i (PQ) = ∂ X i P × (1 ⊗ Q) + (P ⊗ 1) × ∂ X i Q (4)
as well as ∂ X i X j = 1 j=i 1 ⊗ 1 . The fact that the difference quotient takes values in the tensor product allows to keep track of the position of the variable which was differentiated. Moreover, such a definition is a natural extension of classical derivatives applied to matrices. Indeed, if X 1 , . . . , X d are N × N Hermitian matrices, a straightforward computation shows that for all i ∈ {1, . . . , d}, , k, r, s ∈ {1, . . . , N},

∂ X i k (P(X 1 , . . . , X d )) r,s = ((∂ X i P)(X 1 , . . . , X d )) r ,ks , (5)
where (A⊗B) r ,ks := A r B ks . Note that the free difference quotient is not the standard derivative even in the case where one considers only one variable as it is given by

∂ X X k = k-1 =0 X ⊗ X k--1
Identifying the tensor product space with the space of functions of two variables x and y we have

∂ X X k (x, y) = k-1 =0 x y k--1 = x k -y k x -y
and therefore for a general C 1 function we get

∂ X f (x, y) = f (x) -f (y) x -y . ( 6
)
Another derivative of interest is the cyclic derivative D X i given by

D X i P = (m • ∂ X i )P, m(A ⊗ B) = BA .
Again this is a natural notion when applied to matrices as for all i ∈ {1, . . . , d}, k, ∈ {1, . . . , N}

∂ X i k Tr(P(X 1 , . . . , X d )) = (D X i P) k .

Non-commutative laws

A non-commutative law is a linear form µ on C X 1 , . . . , X d with values in C which satisfies 1. The positivity property: for all polynomial P ∈ C X 1 , . . . , X d ) µ(P * ) = µ(P), µ(PP * ) ≥ 0 .

2. The mass condition µ(1) = 1 .

3. The trace property : for all polynomials P, Q ∈ C X 1 , . . . , X d )

µ(PQ) = µ(QP) .
Typical examples are again provided by matrices: if

X N = (X N 1 , . . . , X N d ) are d N × N Hermitian matrices, τ X N (P) := 1 N Tr(P(X N 1 , . . . , X N d ))
is a non-commutative law, with Tr(A) = N i=1 A ii . We will call it the empirical non-commutative law of X N . If X N is random, E[τ X N ] is also a non-commutative law. The non-commutative laws that we shall consider will be bounded, that is that there exists R < ∞ such that for all

i k ∈ {1, . . . , d}, all ∈ N, |µ(X i 1 • • • X i )| ≤ R .
By the Gelfand-Naimark-Segal construction, we can associate to µ a Hilbert space H, Ω ∈ H, and a 1 , . . . , a d bounded linear operators on H so that for all polynomial P µ(P(X 1 , . . . , X d )) = Ω, P(a 1 , . . . , a d )Ω H .

The C * -algebra associated to µ is also described by the C * -algebra generated by a 1 , . . . , a d inside B(H), the set of bounded linear operators on H. We can also associate to µ the von Neumann algebra generated by a 1 , . . . , a d as the weak closure of its C * -algebra in B(H).

Schwinger-Dyson equation

A linear form µ on C X 1 , . . . , X d satisfies the Schwinger-Dyson equation with conjugate variables J = (J 1 , . . . , J d ) iff for any polynomial P,

any i ∈ {1, . . . , d}, µ ⊗ µ(∂ X i P) = µ(J i P)
and µ(1) = 1. This equation makes sense when the J i are polynomials, but also in the C * or the von Neumann algebras associated with µ. We shall however mainly consider this equation when the conjugate variables are polynomials, or absolutely converging series.

We next discuss settings where this equation appears, and start with the enumeration of planar maps.

Generating function for planar maps

We shall see here that generating functions for planar maps satisfy Schwinger-Dyson equation. We summarize below the enumeration problem; more details and discussions can be found for instance in [START_REF] Zvonkin | Matrix integrals and map enumeration: an accessible introduction[END_REF]. A map is a connected graph which is properly embedded into a surface, that is so that its edges do not cross and the faces (obtained by cutting the surface along the edges of the graph) are homeomorphic to disks. The genus g of a map is the genus of this surface. It can be computed thanks to Euler formula

2 -2g = #{vertices} -#{edges} + #{ f aces} . 1 2 3 2 -2g = 2 -3 + 3 1 2 -2g = 2 -3 + 1
We are interested in enumerating maps with a given genus and a given set of vertices. To do that, we shall consider rooted maps, that is maps equipped with a distinguished edge, called the root. This in particular means that we will count maps regardless of some symmetries. In fact, we will even count maps built over sets of vertices which are already drawn on the (orientable) surface and with prescribed degree. A nice way to do that is to start by drawing the vertices on the surface as follows: a vertex of degree p is drawn on an orientable surface together with p "half-edges" around it as well as a half-edge root. In other words, a vertex comes with labelled half-edges corresponding with the way it should be drawn on the surface. The question of enumerating maps then becomes the question of matching the end points of the half-edges. More generally, we will be interested by counting maps with colored edges. For instance, we would like to count maps with n red vertices of degree 4, m blue vertices of degree 4 and p edges between the blue and red vertices: this is called the Ising model on random graphs. To formulate this enumeration problem, we associate (bijectively) colored vertices to monomials in d indeterminates (if we are dealing with enumerating maps with d different colors) as follows. Let q(X 1 , . . . , X d ) = X i 1 X i 2 • • • X i p be a monomial. A "star of type q" is the vertex with first (the root) half-edge of color i 1 , the second color i 2 etc until the last half edge which has color i p .

We then let M((q i , k i ) 1≤i≤m ; g) denote the number of maps with genus g build on k i stars of type q i , 1 ≤ i ≤ m.

Ising model on random graphs

Given monomials q, q 1 , . . . , q k , complex variables t i ∈ C, 1 ≤ i ≤ d, and monomial q, we set

M t (q) = n 1 ,...,n k ≥0 t n i i n i ! M((q, 1), (q i , n i ) 1≤i≤k ; 0)
be the generating function for the enumeration of planar maps with one star of type q (which varies and play in combinatorics the role of an external face) and stars of type q i , 1 ≤ i ≤ k, coming from the model. This formal series can be seen to be absolutely convergent for |t i | ≤ ε, 1 ≤ i ≤ d, for ε small enough. For polynomials P = α i q i , M t (P) is defined by linearity by M t (P) = α i M t (q i ).

Theorem 1. (Tutte [START_REF] Tutte | A census of planar maps[END_REF]) For t 1 , . . . ,t k small enough, M t satisfies the Schwinger-Dyson equation: for all monomial q, all i ∈ {1, . . . , d},

M t ⊗ M t (∂ X i q) = M t (qD i V t ) with V t = 1 2 X 2 i -t i q i .
The proof of this identity follows from the techniques of "surgery" introduced by Tutte. Let us illustrate it in the case of the enumeration of planar maps with vertices of degree 3. Let M(p, n) = M((x n , 1), (x 3 , p); 0) be the number of planar maps with p vertices of degree 3 and one of degree n (again vertices should be thought as stars). Then, we can get an induction relation on M(p, n) by considering what can happen to the root edge of the vertex of degree n (sometimes called the external face). Either it should be matched with an edge of another star (of degree 3) or to another edge of the vertex of degree n.

M(p, n)

In both cases, erasing the matched edge, we arrive either at the situation where the external face has one edge extra, but one star of degree 3 disappeared, or the planar map is cut into two disjoint planar maps (where one contains a number ∈ [0, p] of vertices of degree three, and the other p -). This yields the relation

M(p, n) = 3pM(p -1, n + 1) + n-2 k=0 p =0 p M( , k)M(p -, n -k -2)
Here the combinatorial factors 3p and p are due to the fact that we consider labelled edges; no symmetries are taken into account. Putting M t (x n ) := p≥0 t p p! M(p, n), we deduce from the above that it satisfies the Schwinger-Dyson equation with potential V t = 1 2 x 2tx 3 :

M t ⊗ M t (∂ x x n ) = M t (x n (x -3tx 2 )) .

Non-commutative variables

In the framework of free probability, a non-commutative law µ is said to have conjugate variables J i , 1 ≤ i ≤ d, iff it satisfies the Schwinger-Dyson equation: for any polynomial P, any i ∈ {1, . . . , d} µ ⊗ µ(∂ X i P) = µ(PJ i ) .

Schwinger -Dyson equation can be seen as an integration by part formula: in fact we shall see that it can be retrieved from classical integration by parts when applied to random matrices. In the classical case where the difference quotient would be replaced by the classical derivative and µ simply be a probability measure, the J i would simply be the derivative of the log-density.

The case where J i is a cyclic gradient of a polynomial corresponds to the Schwinger-Dyson equation satisfied by the generating functions for the enumeration of planar maps. In fact, it was shown by D. Voiculescu [START_REF] Voiculescu | A note on cyclic gradients[END_REF] that if the J i are polynomial functions they have to be cyclic gradient of a polynomial. Hence, in this case, the Schwinger-Dyson equations which are relevant are of the same type as those encountered in the enumeration of planar maps. However, in general, potentials need not be small perturbations of the quadratic potential.

Approximate Schwinger-Dyson equations and large random matrices

In this section, we show that large random matrices ensembles satisfy the Schwinger-Dyson equation in average. The first example is given by the Gaussian Unitary Ensemble (GUE) which is described by the N × N Hermitian matrices X N with independent entries which are complex centered Gaussian variables with covariance N -1 above the diagonal and real centered Gaussian variables with covariance N -1 on the diagonal. In other words, the law of X N is given by the law on N × N Hermitian matrices given by

dP N (X N ) = 1 Z N exp{- N 2 Tr((X N ) 2 )}dX N
where dX N = i< j dℜ(X N (i j)) i≤ j dℑ(X N (i j)) is the Lebesgue measure on the entries and Z N is the normalization constant. Take X N 1 , . . . , X N d be d independent GUE matrices. According to the Gaussian integration by parts

(x + iy) f (x + iy, x -iy)e -N(x 2 +y 2 ) dxdy = 1 N ∂ 2 f (x + iy, x -iy)e -N(x 2 +y 2 ) dxdy, (7)
and by definition [START_REF] Biane | Free diffusions, free entropy and free Fisher information[END_REF], one readily finds that

E P ⊗d N 1 N Tr(X N i P(X N 1 , . . . , X N d )) = E P ⊗d N 1 N Tr ⊗ 1 N Tr (∂ X i P)(X N 1 , . . . , X N d )) . ( 8
)
Here and after, E P denotes the expectation under a probability measure P. Hence, independent GUE matrices satisfy the Schwinger-Dyson equation in average, for the quadratic potential V = 1 2 X 2 i . We shall see later that as 1 N Tr(P(X N 1 , . . . , X N d )) self-averages, it converges as N goes to infinity and its limit satisfies the Schwinger-Dyson equation. This can be generalized to the case of random matrices in interaction. To this end take a polynomial V so that

dP V N (X N 1 , . . . , X N d ) = 1 Z V N exp{-NTr(V (X N 1 , . . . , X N d ))}dX N 1 • • • dX N d
makes sense (that is Z V N is well defined and finite). Then, again by integration by parts, one easily sees that the following averaged Schwinger-Dyson equation holds :

E P V N 1 N Tr ⊗ 1 N Tr(∂ X i P) = E P V N 1 N Tr(D X i V P) .
In the case where Z V N is infinite, one can add a cutoff so that the matrices keep an operator norm bounded (this however creates boundary terms in the Schwinger-Dyson equation).

Approximate Schwinger-Dyson equation and Coulomb gas interacting particle systems

Let us consider the following β-models:

dP V,β N (λ 1 , . . . , λ N ) = 1 Z V N i< j |λ i -λ j | β e -N V (λ i ) dλ i
When β = 2, this corresponds to the joint law of the eigenvalues of the matrix models we just considered but with only one matrix. Then, the empirical measure L N = 1 N N i=1 δ λ i satisfies the approximate Schwinger-Dyson equation:

β 2 ∂ X f (x, y)dL N (x)dL N (y) -V (x) f (x)dL N (x) dP V,β N = 1 N ( β 2 -1) f (x)dL N (x)dP V,β N , (9)
for any C 1 function f . Here, ∂ X was defined in [START_REF] Borodin | Gaussian asymptotics of discrete β-ensembles[END_REF]. In this case, we see that an additional error term appears when we deal with the case β = 2: it will yield an expansion of the observable

x k dL N (x)dP V,β N in 1/N instead of 1/N 2 .

Generalization of Schwinger-Dyson equations

It is possible to define analogous equations in other settings, for instance in models build on a structure of planar maps, such as planar algebras [START_REF] Guionnet | Loop models, random matrices and planar algebras[END_REF]. A simple example of planar algebra is given by Temperly-Lieb algebra. Temperley-Lieb elements are boxes containing an even number of boundary points, a starting point, and non-intersecting strings between these points :

*

These elements could as well be endowed with a shading: we do not discuss this point here. We can endow this set with the multiplication:

I := {2 cos( π n )} n≥4 ∪]2
, ∞[, it was shown [START_REF] Guionnet | Random matrices, free probability, planar algebras and subfactors[END_REF] that τ is a tracial state, as a limit of matrix models. This can also be shown by combinatorial arguments [START_REF] Jones | An orthogonal approach to the subfactor of a planar algebra[END_REF]. Moreover, the von Neumann algebra associated to τ was shown to be a factor (and a tower with index δ 2 can be built). τ can also be constructed by matricial approximation. In [START_REF] Guionnet | Loop models, random matrices and planar algebras[END_REF], perturbations of such a law were introduced, allowing to replace the sum over Temperley-Lieb diagrams by a sum over planar diagrams constructed by matching the end points of copies of some Temperley-Lieb elements. This in particular includes the Potts model on random graphs. In these cases, the Schwinger-Dyson equation has a similar graphical interpretation that we gave for planar maps, and can be seen to be equivalent to the following diagrammatic equation:

Tr = i odd Tr Tr i ( 10
)
+ i even Tr W i
Namely, expected values can be seen to amount to either split the diagrams into two disjoint pieces or to glue it with elements from the potential: this is very similar to what we saw for the generating functions of planar maps.

Schwinger-Dyson equation can also be derived for more general matrices, for instance Unitary matrices following the Haar measure [START_REF] Collins | Asymptotics of unitary and orthogonal matrix integrals[END_REF][START_REF] Guionnet | Asymptotics of unitary multimatrix models: The schwingerdyson lattice and topological recursion[END_REF]. Let us for instance consider

P V N (dU 1 , . . . , dU m ) = 1 I N (V, A i ) e NTr(V (U i ,U * i ,A i ,1≤i≤m)) dU 1 • • • dU m for some polynomial V in (U i ,U * i , A i ) 1≤i≤m
and dU the Haar measure on the unitary group. Traces of polynomials also satisfy an averaged Schwinger-Dyson equation but the free difference quotient has to be modified. Let us first define the analogue to the free difference quotient given by the linear maps ∂ j such that for all i, j ∈ {1, . . . , m}

∂ j A i = 0 ∂ j U i = 1 i= j U j ⊗ 1 ∂ j U * i = -1 i= j 1 ⊗U * j ( 11
)
and satisfying the Leibnitz rule (4). We let D j = m • ∂ j be the associated cyclic gradient (recall that m(A ⊗ B) = BA). Using the invariance by multiplication of the Haar measure (which is analogous to using the invariance by translation of Lebesgue measure which led us to the integration by parts we used for Hermitian matrices) one can prove the Schwinger-Dyson equation:

E P V N 1 N Tr ⊗ 1 N Tr(∂ j P) + E P V N 1 N Tr(PD j V ) = 0 . ( 12
)
This is proved by noticing that if we set U j (t) = U j e itB and leave the other U k (t) = U k unchanged for a Hermitian matrix B then for all k, l ∈ {1, . . . , N}

∂ t P(U p (t), 1 ≤ p ≤ m, A i )(k, l)e NTr(V (U p (t),U p (t) * ,A i ) dU 1 • • • dU m = 0
Taking B to be the matrix B st = 1 st=kl +1 st=lk or B st = i1 st=kl -i1 st=lk shows that we can choose by linearity B to be the matrix B st = 1 st=kl even though this is not self-adjoint. This yields the result after summation over k and .

It is also possible to define Schwinger-Dyson equations in case of non tracial states [START_REF] Nelson | Free monotone transport without a trace[END_REF], that is type III factors. Again the appropriate differential calculus has to be developed. Another extension concerns discrete analogue of the β-ensembles consisting in summing over λ i taking values in a lattice instead of on the real line. Integration by parts give equations which are not easy to handle, but Nekrasov [START_REF] Nekrasov | Quantum geometry and quiver gauge theories[END_REF][START_REF] Nekrasov | Seiberg-witten geometry of four dimensional n = 2 quiver gauge theories[END_REF][START_REF] Nekrasov | Non-perturbative dyson-schwinger equations and bps/cft correspondence[END_REF] defined non-perturbative Schwinger-Dyson equations which can be analyzed [START_REF] Borodin | Gaussian asymptotics of discrete β-ensembles[END_REF].

To sum up, the Schwinger-Dyson equations show up in many non-commutative or ordered systems. The free difference quotient appears as the natural derivation in this setting. We shall now analyze these equations.

Properties of solutions to Schwinger-Dyson equations

Even though the Schwinger-Dyson equation can be seen as an integration by part formula in free probability, it is not clear at all that it has a unique solution. In fact it does not in general. Indeed, even if d = 1, there may be several solutions to the Schwinger-Dyson equation. Indeed, we can choose V so that any solution will have the same disconnected support, for instance by taking V with two global minima and with large enough values in between these minima. Then, the Schwinger-Dyson equation will have a solution for any choice of mass given to these connected components. We start by investigating this question of uniqueness and then we study properties of the solutions when they are unique.

Uniqueness

Let µ be a linear form on

C X 1 , • • • , X d so that µ ⊗ µ(∂ X i P) = µ(D X i V P) ∀P ∈ C X 1 , . . . , X d , ∀i ∈ {1, . . . , d} . (13)
Assume µ(1) = 1. We investigate in the next subsection the cases where we know that a unique solution exists.

Quadratic case

Note that when V = 1 2 X 2 i , uniqueness is clear since then

µ(X i P) = µ ⊗ µ(∂ X i P) allows to define µ(X i 1 • • • X i k )
for all choices of i 1 , . . . , i k ∈ {1, . . . , d} by induction over k, starting from µ(I) = 1. This solution is denoted by σ d . It is the solution for the enumeration of maps with one star and therefore we see that

σ d (X i 1 • • • X i k )
corresponds to the number of planar maps that one can build over a star of type X i 1 • • • X i k . Note that we can equivalently see a star as a circle decorated with colored dots one colored dot (with color i 1 ) and with colored dots drawn in the clockwise order of color i 2 , i 3 , . . . , i k , and that in case of a single star this is also equivalent to a line with first point with color i 1 , second i 2 , . . . In the later case, the planar map can be identified with a non-crossing pair partition blocks of points of the same color. The non-crossing property is simply that if (a, b) and (c, d) are two pairings then a < b < c < d or a < c < d < b and i a = i b , i c = i d .This interpretation of σ d (q) can be compared with that of Gaussian moments which are given by the enumeration of pair partitions. In fact, σ d plays the role of the law of independent Gaussian variables in free probability. We will see that it also gives the limit of moments of GUE variables. Moreover, it encapsulates the notion of freeness which is central in free probability. In fact, σ d can be defined as follows:

• if we restrict σ d to polynomials in one variable, it is given by the semi-circle distribution, that is

σ d (X k i ) = 1 2π 2 -2
x k 4x 2 dx .

• whereas more general moments can be computed from the property that X 1 , . . . , X d are free under σ d : we say that X 1 , . . . , X d are free under µ iff

µ(q 1 (X i 1 )q 2 (X i 2 ) • • • q (X i )) = 0 ( 14
)
for any polynomials q k , 1 ≤ k ≤ , so that µ(q k (X i k )) vanishes and any indices i j ∈ {1, . . . , d} so that i k = i k+1 , 1 ≤ k ≤ -1. It is not hard to see that ( 14) defines uniquely the law once the marginal distributions are known.

The equivalence of the two definitions of σ d above, by planar maps or freeness, amounts to showing that any planar map built on a star of type q must leave at least one subset of connected one-color half-edges (corresponding to each of the monomials q i ) to having only selfmatchings (this contribution then vanishes by centering). This can be checked by induction over the degree of q.

3.1.2 Perturbative case Theorem 2.

[25] Let R ∈ (2, +∞) and V = 1 2 X 2 i + εW for a self-adjoint polynomial W = p i=1 t i q i , for some monomials q i and complex numbers t i . Then for ε small enough (depending on R) there exists a unique solution µ V to (13) so that µ(1) = 1 and for any k ∈ N, any i 1 , . . . ,

i k ∈ {1, . . . , d} k |µ(X i 1 • • • X i k )| ≤ R k . ( 15
)
Moreover it is a tracial state: for any polynomials P, Q,

µ(P * ) = µ(P), µ(PP * ) ≥ 0, µ(PQ) = µ(QP) .
The idea of the proof is perturbative, the situation t i ≡ 0 being clear according to the previous section. Indeed, take two solutions τ, τ and denote by

∆ k := sup q:deg(q)≤k |τ(q) -τ(q)|
where the supremum is taken on monomials (or words) of degree smaller or equal to k. We have

∆ k+1 = max i sup q:deg q=k |τ(X i q) -τ(X i q)|
whereas by using ( 13), [START_REF] Collins | Asymptotics of unitary and orthogonal matrix integrals[END_REF] and ∆ 0 = 0, if D + 1 = max 1≤i≤p deg(q i ) we get for any monomial q with degree k,

|τ(X i q) -τ(X i q)| ≤ |τ ⊗ τ(∂ X i q) -τ ⊗ τ(∂ X i q)| + Dε p j=1 |t j |∆ k+D-1 ≤ 2 k-1 l=1 ∆ l R k-1-l + Dε p j=1 |t j |∆ k+D-1 Hence, ∆ k+1 ≤ 2 k-1 l=1 ∆ l R k-1-l + Dε |t j |∆ k+D-1 , ∆ k ≤ 2R k so that for γ < R -1 ∧ 1, we deduce that ∆ γ := k≥1 γ k ∆ k ≤ 2γ 2 1 -Rγ ∆ γ + εD |t j | γ D-2 ∆ γ . (16)
Hence, if ε is small enough so that there exists γ ∈ (0, 1/R ∧ 1) such that 2γ 2 1 -Rγ + εD |t j | γ D-2 < 1 we deduce that ∆ γ = 0, which implies τ = τ.

Strictly convex case

We need to find the right definition of strict convexity in the non-commutative setup. It should as in the classical case be related with the idea that a second order differential of the function is bounded below by a multiple of the identity. The difficulty here is that derivatives have values in the tensor product, and there are many possible topologies that can be implemented on the space of tensor product valued functions. We will use classes of differentiable functions with values in the Haagerup tensor product M ⊗ ehcD 2 of some von Neumann algebra M. We do not detail more precisely this space here (see e.g. [START_REF] Dabrowski | Analytic functions relative to a covariance map η: I. generalized haagerup products and analytic relations[END_REF]) but emphasize that it is nice, for instance to make sure that composition of smooth functions are smooth, or to get smoothness of natural operations that show up in the analysis. Proposition 3. The following are equivalent.

1. A = A * ∈ M d (M ⊗ ehcD 2
) has a semigroup of contraction e -At for all t ≥ 0.

A

= A * ∈ M d (M ⊗ ehcD 2 ) has a resolvent family for all α > 0, α+A is invertible in M d (M ⊗ ehcD 2 )
and || α α+A || ≤ 1. In this case we say A ≥ 0. Definition 4. Let c, R ∈ R +, * . Let V = V * be a smooth function on non-commutative variables.

V is said to be generalized (c, R)-convex if (∂ X i D X j V )(X 1 , . . . , X d ) -cId ≥ 0 for some c > 0 and all X 1 , . . . , X d self-adjoint variables bounded by R in the von Neumann algebra M.

We then can prove that Theorem 5. Let R > 0. Then there exists c(R) > 0 so that if V is (c, R)-convex for some c ≥ c(R), there exists a unique solution to the Schwinger-Dyson equation [START_REF] Chekhov | Matrix eigenvalue model: Feynman graph technique for all genera[END_REF].

The arguments to prove this theorem are more sophisticated than the previous ones as they appeal to dynamics. In fact, it is shown that the solutions to Schwinger-Dyson equation are stationary solutions of the free stochastic differential equation given by

X i t = X i + S i t - 1 2 t 0 D X i V (X s )ds (17)
where (S 1 , S 2 , . . . , S d ) are d free free Brownian motion. Such equations are well defined and discussed in [START_REF] Biane | Stochastic calculus with respect to free Brownian motion and analysis on Wigner space[END_REF][START_REF] Biane | Free diffusions, free entropy and free Fisher information[END_REF]. Let us shortly describe them. The free Brownian motion S is constructed similarly to the Brownian motion by the requirements that 1. S 0 = 0, 2. For any

t 1 ≤ t 2 ≤ • • • ≤ t r , (t r -t r-1 ) -1/2 (S t r -S t r-1 ), . . . , (t 2 -t 1 ) -1/2 (S t 2 -S t 1 ),t -1/2 1 S t 1
are free semi-circular variables with distribution σ r .

(S 1 , S 2 , . . . , S d ) are free in the sense that their marginals satisfy [START_REF] Collins | Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability[END_REF]. Existence and uniqueness of solutions to ( 17) can be proved by using standard Picard arguments (here we assume that V is a polynomial, but any non-commutative function satisfying a Lipschitz property would work) and the fact that solutions stay bounded uniformly in time by R if their initial condition has sufficiently small norm when the potential is (c, R)-convex, see [START_REF] Biane | Stochastic calculus with respect to free Brownian motion and analysis on Wigner space[END_REF]. Let τ be the non commutative law of X t = (X 1 t , . . . , X d t ),t ≥ 0, with initial condition X. The free analogue of Itô's calculus implies that the distribution of X t satisfies the differential equation

τ(P(X t )) = τ(P(X)) + t 0 τ ⊗ τ( 1 2 d i=1 ∂ X i D X i P(X s ))ds - 1 2 t 0 τ( d i=1 D X i P(X s )D X i V (X s ))ds .
Hence, we see that the right hand side vanishes when the non-commutative distribution of X s satisfies the Schwinger-Dyson equation. Moreover, the strict convexity of V allows to show, as in the classical case, that the law of X s does not depend much on the initial condition X when s goes to infinity, hence implying uniqueness of the stationary measure. This allows to conclude.

Remark 6. Uniqueness of solutions fails in general, even in the case d = 1 as we already noticed.

Free transport and isomorphisms

In this section, we discuss the construction of transport maps between non-commutative laws. Let τ, µ be two non-commutative laws of d (resp. m) variables X = (X 1 , . . . , X d ) (resp. Y = (Y 1 , . . . ,Y m )). We seek for "transport maps" T = (T 1 , . . . , T m ) and T = (T 1 , . . . , T d ) of d (resp. m) variables so that for all polynomials P, Q

τ(P(X)) = µ(P(T 1 (Y ), . . . , T d (Y ))) µ(Q(Y )) = τ(Q(T 1 (X), . . . , T m (X)))
In this case, we denote τ = T µ and µ = T τ. At this point, transport map could either be polynomials, or in the C * algebra, or in the von Neumann algebra associated with the noncommutative laws by the GNS construction. In fact, the existence of such transport maps gives isomorphisms between these algebras (depending on the regularity of the transport maps). The classical analogue of such a question is, for given two probability measures µ and ν on R d and R m respectively, whether we can build a transport map from µ to ν, that is a measurable function T : R d → R m so that for all bounded continuous function f f (T (x))dµ(x) = f (x)dν(x) .

We denote T #µ = ν. We can similarly seek for T so that T #ν = µ. Von Neumann proved that, except for very degenerate cases (such as µ being a Dirac mass at a point), one can always build a transport map between two probability measures. For instance, one can always build transport maps from the Lebesgue measure on [0, 1] to any probability measure on R m for any m ≥ 1. In fact, this transport map may be quite rough, and no than measurable.

In the non-commutative setting, this type of question is widely open. In fact, a result of Ozawa [START_REF] Ozawa | There is no separable universal II 1 -factor[END_REF] shows that there is no non-commutative law analogous to Lebesgue measure in the sense that any non-commutative measure could be seen as its image by some transport map.

In this section, we want to discuss the existence of such transport map in very nice cases, in fact for solutions to Schwinger-Dyson equation. We will restrict ourselves to when this equation has a unique solution as seen in the previous section. So let V be a self-adjoint polynomial and consider the solution µ V to the Schwinger-Dyson equation

µ V ⊗ µ V (∂ X i P) = µ V (D X i V P) . (18)
When we restrict ourselves to pertubative setting, a natural topology on the space of functions is given by the Banach norm

P R = | q, P |R deg(q) ( 19
)
where the sum is over monomials q entering in the decomposition of P by P = q, P q and deg(q) denotes the degree of the monomial q. We let A R be the completion of the set of polynomials for . R .

Theorem 7 (G.-Shlyakhtenko [START_REF] Guionnet | Free monotone transport[END_REF]). Let A > A > 4 and V = 1 2

X 2 i + βW self-adjoint with W A < ∞.
Then, for β small enough, there exist F V , T V transport maps in A A between µ V and

σ d = µ X 2 i /2 so that µ V = T V σ d , σ d = F V µ V .
In particular the related C * algebras and von Neumann algebras are isomorphic.

In a work in progress with Y. Dabrowski, we extend this result to the strictly convex case:

Theorem 8 (Dabrowski-G-Shlyakhtenko [START_REF] Dabrowski | Free transport for convex potentials[END_REF]). Let R > 4. Then, there exists c(R) > 0 finite so that for any (c, R) function V , the C * and W * -algebras associated with the unique solution µ V to [START_REF] Dabrowski | Free transport for convex potentials[END_REF] are isomorphic to that of d free semicircular variables.

These results allow to prove new isomorphisms. Let us for instance consider q-Gaussian laws introduced by Bozejko and Speicher [START_REF] Biane | Stochastic calculus with respect to free Brownian motion and analysis on Wigner space[END_REF] as an interpolation between classical Gaussian variables and free semi-circular laws. The moments of d q-Gaussian variables are described by

τ q,d (X i 1 • • • X i p ) = π q i(π) ∀i k ∈ {1, • • • , d}
where the sum is over pair partitions and i(π) counts the number of crossings. Hence, q = 1 corresponds to commutative Gaussian variables whereas the case q = 0 corresponds to free semi-circular variables. It was shown by Y. Dabrowski that for qd small enough, τ q,d satisfies the Schwinger-Dyson equation with a potential V which is a small perturbation of the quadratic potential in some A R , as in Theorem 7. Hence, we can apply our result to prove the following corollary.

Corollary 9.

[29] Let d ∈ N * . For qd small enough, there exist R > 0 finite and transport maps in A R between τ q,d and τ 0,d = σ d . In particular the C * -algebra and von Neumann algebras of q-Gaussian laws, q small, are isomorphic to that of the free semicircle law σ d .

The ideas to construct such transport maps follow very classical ideas going back to Monge-Ampère, that we now describe in the classical probability setting. Let µ, ν be probability measures on R d that have smooth densities

µ(dx) = e -V (x) dx, ν(dx) = e -W (x) dx .
Then T #µ = ν is equivalent to

f (T (x))e -V (x) dx = f (x)e -W (x) dx = f (T (y))e -W (T (y)) JT (y)dy
with JT the Jacobian of T . Hence, it is equivalent to the transport equation

V (x) = W (T (x)) -log JT (x) .
Monge-Ampère equation amounts to taking T to be a gradient. Such an equation can not be solved in general, except in the case where V -W is small. Indeed, then we can look for a solution T close to identity, in which case JT is close to one and therefore the equation does not touch the singularity of the logarithm. Hence, it can be solved by the implicit function theorem. This type of arguments can be generalized to the non-commutative setting by showing that the Schwinger-Dyson equation is equivalent to a free Monge-Ampère equation. The latter where above we have used conditional expectation with respect to the initial data. The above integral converges absolutely provided that we have a strictly convex interaction. The main point to finally construct the transport by the transport equation is to make sure that ψ t described above is a sufficiently smooth function. This in fact requires to understand better the set up of free analysis and its natural topologies.

Models asymptotically driven by ∂ X

In this section we investigate models which are only asymptotically driven by the free difference quotient, such as random matrices or particles in Coulomb gas interaction. We show that the approximate Schwinger-Dyson equation then allows to get a topological expansion, that is an expansion of observables in terms of the dimension. The first order is given by the solution to the Schwinger-Dyson equation, and the corrections are computed recursively by linearizing the approximate Schwinger-Dyson equation around their limit.

This expansion is said to be topological because if the dimension is seen as a parameter, then the observables can be seen as generating functions for maps of all genus. This can be proved again because the equations are driven by the free difference quotient, yielding a combinatorial interpretation: each correction can be interpreted as some number of maps with a given genus. We also investigate transport maps in this setting and show that we can construct approximate transport maps to prove this time the universality of the local fluctuations of the eigenvalues. Hence, this section follows roughly the scheme we have developed in the previous section, but pushed to get corrections of the solutions of the approximate Schwinger-Dyson equation with respect to the solution of the limiting equation.

Matrix models

Let V be a self-adjoint polynomial in d non-commutative indeterminates and consider the matrix integral

dP V N (X N 1 , . . . , X N d ) = 1 Z V N e -NTr(V (X N 1 ,...,X N d )) i 1 X N i ≤R dX N i ( 22
)
where X denotes the spectral radius of the matrix X. dX N i denotes the Lebesgue measure. We denote in short Tr(q) for Tr(q(X N 1 , . . . , X N d )) and dP V N for dP V N (X N 1 , . . . , X N d ). Then we first state the topological expansion for this matrix model.

Theorem 10 (G-Maurel-Segal [START_REF] Guionnet | Combinatorial aspects of matrix models[END_REF]). Take R ∈ (2, +∞) and fix K ∈ N. Let q i , 1 ≤ i ≤ p, be monomials. Assume V t = V * t = 1 2 X 2 i -p i=1 t i q i with t i small enough. Then 1.

E

P V t N [ 1 N Tr(q)] = K g=0 1 N 2g τ t g (q) + o( 1 N 2K ) 22 with τ t g (q) = k i ≥0 t k i i
k i ! M((q, 1), (q i , k i ) 1≤i≤p ; g) where M((q, 1), (q i , k i ) 1≤i≤p ; g) is the number of maps with genus g build over a star of type q and k i stars of type q i , 1 ≤ i ≤ p.

2.

1

N 2 log Z N V t Z N V 0 = K g=0 1 N 2g F g (t) + o( 1 N 2K )
where F g (t) =

t k i i k i ! M((q i , k i ) 1≤i≤p ; g).
Note that the cutoff R is mainly required to make the partition function Z N V finite. In the case when V is strictly convex, it can be removed. The main tool to find these asymptotic expansions is again the Schwinger-Dyson equation which is a consequence of integration by parts and given by

E P V N 1 N Tr(D X i (V )P) = E P V N 1 N Tr ⊗ 1 N Tr(∂ X i P) . (23)
Moreover, to study these equations we will need tools such as concentration of measure and Brascamp-Lieb inequality (see [START_REF] Anderson | An introduction to random matrices[END_REF] and [START_REF] Guionnet | Large random matrices: lectures on macroscopic asymptotics[END_REF]). They imply that there exists a finite constant C(which only depends on c) so that for any monomial q of degree less than

√ N | 1 N Tr(q)|dP V N ≤ C deg q , (24) and 1
N Tr(q) - 1 N Tr(q)dP V N 2 dP V N ≤ C deg(q) N 2 . ( 25
)
As a consequence of (24), the family { 1 N Tr(q)dP V N , q} indexed by monomials in noncommutative variables is tight. Any limit point {τ(q), q} can be extended by linearity to polynomials and then satisfies the Schwinger-Dyson equation

µ V (PD X i V ) = µ V ⊗ µ V (∂ X i P) (26)
with µ V (I) = 1. Moreover, for any monomial q, we deduce from (24) that

|µ V (q)| ≤ C deg(q) . (27)
But we know by Theorem 2 that such an equation has a unique solution provided that the t i 's are small enough. We then deduce readily that { 1 N Tr(q)dP V N , q} converges towards {µ V (q), q} with µ V the unique solution to the Schwinger-Dyson equation [START_REF] Guionnet | Second order asymptotics for matrix models[END_REF].

To derive the corrections to this limit, the first point is to prove an a priori rough estimate by showing that there exists a finite constant C > 0 so that for all t i 's small enough, all monomials q of degree less than N 1/2-ε for ε > 0, we have

|E P V N [ 1 N Tr(q) -µ V (q)| ≤ C deg(q) N 2 . ( 28
)
The proof elaborates on the ideas developed around [START_REF] Dabrowski | A note about proving non-Γ under a finite non-microstates free Fisher information assumption[END_REF] to prove uniqueness of the solution to Schwinger-Dyson equation and the concentration inequalities [START_REF] Guionnet | Combinatorial aspects of matrix models[END_REF], see [START_REF] Guionnet | Second order asymptotics for matrix models[END_REF]. We next turn to the precise estimate of the asymptotics of δ N (P) = E P V N [Tr(P)] -Nµ V (P). To this end, we shall introduce the following cumulants:

W V 2 (P, Q) = E P V N [(TrP -E P V N TrP)(TrQ -E P V N [TrQ])] = ∂ ε P V -εN -1 Q (TrP)| ε=0 W V 3 (P, Q, R) = ∂ ε W V -εN -1 R 2 (P, Q) .
Rewriting [START_REF] Guionnet | Random matrices, free probability, planar algebras and subfactors[END_REF] reveals after some algebra:

E P V N [Tr(Ξ i P)] = 1 N W V 2 (∂ X i P) + 1 N δ N ⊗ δ N (∂ X i P) , (29)
where

Ξ i P = ∂ X i V #P -(µ V ⊗ I + I ⊗ µ V )∂ X i P .
By our a priori estimate (28) on δ N the last term in the right hand side of ( 29) is at most of order N -3 . Hence, to estimate the first order correction, we would like to estimate the asymptotics of W V 2 as well as "invert" Ξ i . It turns out that even though Ξ i is hardly invertible, a combination of the Ξ i 's is. Because Ξ can be modified by cyclic rearrangement since it is taken under the trace, we let Ξ be the operator on A 0 = {P ∈ C X 1 , . . . , X d : µ V (P) = 0} given by ΞP

= i (∂ X i P#D X i V -(µ V ⊗ I + I ⊗ µ V )∂ X i D X i P) .
Then the image of Ξ lies in A 0 by the Schwinger-Dyson equations. We see that in the case where V = X 2 i /2, we have for any monomial P

Ξ 0 P := i ∂ X i P#D X i V = deg(P)P
that is Ξ is the sum of the degree operator and an operator which lowers the degree. It can be inverted on the space of polynomials with no constant terms : indeed for any non constant monomial P we have

Ξ -1 0 (P) = 1 deg(P) P .
When V is a small perturbation of the quadratic potential, it can be seen that Ξ is invertible on the space given by the closure of polynomials for the norm . C , see [START_REF] Dabrowski | Unbounded derivations, free dilations, and indecomposability results for II 1 factors[END_REF].

To estimate W V 2 , we obtain a new equation by replacing V by V + εW and identifying the linear terms in ε in [START_REF] Guionnet | Random matrices, free probability, planar algebras and subfactors[END_REF]. we find

W V 2 (P,W ) = E P V N [ 1 N Tr( i D X i Ξ -1 PD X i W )] +N -1 W V 3 ( i ∂ X i D X i Ξ -1 P,W ) + (W V 2 ⊗ δ N + δ N ⊗W V 2 )( i ∂ X i D X i Ξ -1 P,W ) .
It turns out that the terms in W V 3 and W V 2 are bounded by concentration inequalities (25) whereas δ N is bounded by our previous rough estimate [START_REF] Guionnet | Free diffusions and matrix models with strictly convex interaction[END_REF]. Hence we conclude that

lim N→∞ W V 2 (P,W ) = µ V ( i D X i Ξ -1 P × D X i W ) =: w 2 (P,W )
and therefore plugging this back into (29) we deduce the first order correction is given by

E P V N [ 1 N Tr(P)] = µ V (P) + 1 N 2 w 2 [ i ∂ X i D X i (Ξ -1 P)] + o(N -2 ) .
The next orders of the asymptotic expansion can be derived similarly, by considering a family of Schwinger-Dyson equations which are obtained by perturbating the first one with respect to small additional potentials. We refer the interested reader to [START_REF] Guionnet | Second order asymptotics for matrix models[END_REF][START_REF] Maurel-Segala | High order expansion of matrix models and enumeration of maps[END_REF] for full details. Hence, we see that deriving the topological expansion boils down to

• Derive Schwinger-Dyson equations,

• Use concentration arguments to be able to separate elements on different scales,

• Invert a linear differential operator, which can be interpreted as the limit of the infinitesimal generators related with the Gibbs measures.

It turns out that Schwinger-Dyson equations can be established for many other models which are not directly related with Gaussian random matrices. It seems that a large family of Schwinger-Dyson equations give rise to topological expansions. We describe below the case of the βensembles and the integration over the unitary group.

Topological expansion for β-matrix models

The law of the eigenvalues of the GUE follows the distribution on R

N dP N (λ) = 1 Z N i< j |λ i -λ j | 2 e -N λ 2 i dλ i
as can be checked by doing the change of variables associating to X its ordered eigenvalues and a parametrization of its eigenvectors. β-ensembles are the following generalization of this distribution: dP

V,β N (λ) = 1 Z V N,β i< j |λ i -λ j | β e -N V (λ i ) dλ i
It is related with invariant matrix ensembles only in the cases β = 1, 2, 4 and a priori has no relations with Gaussian entries otherwise. However, it was proved in [START_REF] Borot | Asymptotic Expansion of β Matrix Models in the one-cut Regime[END_REF], see [START_REF] Chekhov | Matrix eigenvalue model: Feynman graph technique for all genera[END_REF] for a formal proof, that β-ensembles have a large-N (also called topological) expansion which generalizes the expansion we discussed for GUE matrices in the previous section (restricted to d = 1). More precisely, assume that Assumption 11.

• lim |x| →∞ V (x) 2β log |x| > 1,
• V is analytic in a neighborhood of the real line,

• Consider

I(µ) = V (x) +V (y) 2 - β 2 log |x -y| dµ(x)dµ(y) (30)
It is strictly convex and therefore has a unique minimizer µ V which is characterized by the fact that the effective potential V eff (x) = V (x) -2β log |x -y|dµ V (y) is equal to a constant C on the support of µ V and greater than C outside of the support. We assume that the support of µ V is connected.

• Assume also that the effective potential V eff achieves its minimal value only on the support of µ V .

• The density of µ V vanishes like a square root at the boundary of the support.

Then for any z ∈ C\R, and K ≥ 0

1 N N i=1 1 z -λ i dP V,β N (λ) = K k=0 N -k W V,k (z) + o(N -K ) (31)
where o(N -K ) is uniform on compacts. Moreover, we have

W V,k (z) = k/2 g=0 β 2 -g 1 - 2 β k-2g+1 W V ;(g;k-2g+1)(z) .
Note that the hypothesis that the support is connected is important since otherwise the result is not true in general. The proof of this expansion relies as well on the Schwinger-Dyson equations [START_REF] Borot | Asymptotic Expansion of β Matrix Models in the one-cut Regime[END_REF]. As a consequence, one sees that the equilibrium measure µ V satisfies the limiting Schwinger-Dyson equation

β 2 f (x) -f (y) x -y dµ V (x)dµ V (y) = f (x)V (x)dµ V (x) . ( 32
)
If V is a small perturbation of the quadratic potential one can develop arguments similar to those of the previous section to check that moments under µ V are generating functions for planar maps.

In fact, the limiting equation [START_REF] Maurel-Segala | High order expansion of matrix models and enumeration of maps[END_REF] does not always have a unique solution as it is a weak characterization of the minimizers of ( 30), but it does as soon as the support of the limiting equation has a connected support. Indeed in this case we can use the theory of integral equations, and in particular that of the airfoil equation [START_REF] Tricomi | Integral equations[END_REF], to get uniqueness of the solution of Schwinger-Dyson equation and invertibility of the relevant linear operators. In any case, µ V governs the first order of the expansion. To get the higher order terms in the expansion the idea is, as in the previous section, to write equations for all the cumulants

W V n (x 1 , . . . , x n ) = ∂ ε 1 • • • ∂ ε n ln Z V -2 βN i ε i x i -• N,β ε i =0
by differentiating the Schwinger-Dyson equation ( 9) with respect to the potential. As the linear differential operator Ξ appearing when one linearizes the Schwinger-Dyson equation around its limit is invertible (in the connected support case), the strategy developed in the previous section can be generalized.

In the case where the support of the limiting equilibrium measure has r connected components S j , 1 ≤ j ≤ r, r ≥ 2, it turns out that the Schwinger-Dyson equation [START_REF] Maurel-Segala | High order expansion of matrix models and enumeration of maps[END_REF] has a unique solution given the mass of each connected components S j , 1 ≤ j ≤ r. Moreover, in this case the operator Ξ is as well invertible. Hence, we can use the previous scheme to expand in N the partition function

Z V n 1 ,...,n r = r s=1 n 1 +•••+n j j=n 1 +•••+n j-1 +1 1 λ j ∈S δ j i< j |λ i -λ j | β e -N V (λ i ) dλ i
where S δ j = {x : d(x, S j ) ≤ δ} for some δ > 0. The expansion does not depend on δ because the eigenvalues will stay inside ∪S j with very large probability. This allows to retrieve [START_REF] Borot | Asymptotic Expansion of β Matrix Models in the multi-cut Regime[END_REF] an expansion in N for the partition function where the number of eigenvalues in each S j is randomly chosen:

Z V N = n 1 +•••+n r =N Z V n 1 ,...,n r
and for the correlators W V n (x 1 , . . . , x n ). As the sum above is dominated by one set of filling fractions n 1 , . . . , n r , these expansions contain a Theta-function, representing the fact that the eigenvalues at the boundary of the support will tunnel from S j to S j+1 for some j ∈ {1, . . . , r -1} following a discrete gaussian law (which depends on the values of Nµ V (S j ) -Nµ V (S j ) , 1 ≤ j ≤ r, and therefore may only converge under subsequences). In particular, central limit theorem for linear statistics do not hold in general, see [START_REF] Borot | Asymptotic Expansion of β Matrix Models in the multi-cut Regime[END_REF]Section 8.3] and also [START_REF] Pastur | Limiting laws of linear eigenvalue statistics for Hermitian matrix models[END_REF].

Finally, the previous considerations also generalize [START_REF] Borot | Large-n asymptotic expansion for mean field models with coulomb gas interaction[END_REF] to more general potentials of the form N -k+1 N i 1 ,...,i k =1 V (λ i 1 , . . . , λ i k ). A more complicated generalization is to deal with models which interact via a potential which vanishes at the origin, but which is not homogenous as the Coulomb gas interaction. In a recent work with Borot and Kozlowski, we consider the case where the interaction is given by i< j sinh(ω 1 (λ iλ j )) sinh(ω 2 (λ iλ j )) .

Topological expansion for the Haar measure on the unitary group

In this section we shall consider the Haar measure dU on the unitary group, that is the unique measure on U(N) which is invariant under left multiplication by unitary matrices and with mass one. We consider matrix integrals given by

I N (V, A i ) = e NTr(V (A i ,U i ,U * i ,1≤i≤m)) dU 1 • • • dU m ,
where (A i , 1 ≤ i ≤ m) are N × N deterministic uniformly bounded matrices, and V is a polynomial function in the non-commutative variables (U i ,U * i , A i , 1 ≤ i ≤ m). A well-known example is the Harich-Chandra-Itzykson-Zuber integral HCIZ(A 1 , A 2 ) = e NTr(A 1 UA 2 U * ) dU .

For technical reasons, we assume that the polynomial V satisfies Tr(V (U i ,U * i , A i , 1 ≤ i ≤ m)) ∈ R for all U i ∈ U(N) and all Hermitian matrices A i , 1 ≤ i ≤ m and N ∈ N. Under those very general assumptions, the formal convergence of the integrals could already be deduced from [START_REF] Collins | Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability[END_REF]. The following theorem is a precise description of the results from [START_REF] Collins | Asymptotics of unitary and orthogonal matrix integrals[END_REF][START_REF] Guionnet | Asymptotics of unitary multimatrix models: The schwingerdyson lattice and topological recursion[END_REF] which gives an asymptotic expansion: Theorem 12. [START_REF] Collins | Asymptotics of unitary and orthogonal matrix integrals[END_REF][START_REF] Guionnet | Asymptotics of unitary multimatrix models: The schwingerdyson lattice and topological recursion[END_REF]Let K ∈ N. Under the above hypotheses and if we further assume that the spectral radius of the N × N matrices (A i , 1 ≤ i ≤ m) is uniformly bounded (by say M), there exists ε = ε(M,V ) > 0 so that for z ∈ [-ε, ε], the expansion

  and the trace given by τ(S) = R∈TL δ loops in S.R where the sum holds over Temperley diagrams with the same number of boundary points as S and T T.S= S In the above drawing, T.S has two loops. For δ ∈

1 N 2

 2 log I N (zV, A i ) = , F V,τ (z) is an analytic function of z ∈ {z ∈ C : |z| ≤ ε} which only depends on the empirical non-commutative law of the (A i ) 1≤i≤m . Furthermore, if we letP zV N (dU 1 , . . . , dU m ) = 1 I N (zV, A i ) e zNTr(V (U i ,U * i ,A i ,1≤i≤m)) dU 1 • • • dU m

This work was partially supported by the NSF and Simons foundation

takes the form

where S is a certain symmetrization operator and J is a natural generalization of the notion of Jacobian based on the free difference quotient. Based on this formula, one can develop in the non-commutative setting arguments similar to those of the implicit function theorem.

B. Nelson showed that it is also possible to extend these ideas to more general setups; to construct a transport map in the case of type III factors [START_REF] Nelson | Free monotone transport without a trace[END_REF] and in the case of finite depth sub factor planar algebras [START_REF] Nelson | Free transport for finite depth subfactor planar algebras[END_REF].

Such arguments do not work a priori to prove non-perturbative results and with Dabrowski and Shlyakhtenko, we used a follow up to this idea based on interpolating potentials. Let us first outline this idea in the classical set up. Then, again consider for potentials V,W properly renormalized, the probability measures

Some algebra reveals that Monge-Ampère equation becomes

if we assume that ϕ t = ∇ψ t . Moreover L t is the infinitesimal generator :

Hence, to construct a transport map it is enough to solve the Poisson equation ( 21) and then the transport equation [START_REF] Deift | Random matrix theory: invariant ensembles and universality[END_REF] driven by ∇ψ t to find T 0,t . Taking t = 1 provides a transport map from µ W to µ V . To solve the Poisson equation [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF], one needs to invert L t , that is find the Green function. Alternatively, one can consider the semi-group P t s = e sL t and in the case when it converges fast enough to a unique invariant measure µ t (for instance in the case when V t is strictly convex), use that ( 21) is satisfied if

This scheme can be generalized to the free setting by using again free stochastic differential equations. Indeed, for a strictly convex interaction, if X Y,t . is the solution of the free SDE with potential V t and initial condition Y , we can define

for every polynomial P in (U i ,U * i , A i ) 1≤i≤m we have the expansion

A combinatorial interpretation of the limits F zV 0 and τ zV 0 in terms of planar maps was provided in [START_REF] Collins | Asymptotics of unitary and orthogonal matrix integrals[END_REF], but is still open for higher orders F zV g , τ zV g , g ≥ 1. The strategy of the proof of Theorem 12 is again to study the Schwinger-Dyson equations under the associated Gibbs measure P V N , that is [START_REF] Bourgade | Universality of general β-ensembles[END_REF]. By using concentration of measure, we know that for every polynomial P, N -1 Tr(P(U i ,U * i , A i )) is not far from its expectation and therefore we deduce that the limit points of these (bounded) quantities τ(P) satisfy the Schwinger-Dyson equation

where ∂ j is the modified free difference quotient as given in [START_REF] Bourgade | Edge universality of beta ensembles[END_REF] and

Uniqueness of the solution to such an equation in the perturbative regime is done as in the Gaussian case; when z = 0 it is clear as it defines all moments recursively from the knowledge of the empirical non-commutative law τ A and a perturbation argument shows this is still true for small parameters. Uniqueness of the solutions to [START_REF] Nekrasov | Non-perturbative dyson-schwinger equations and bps/cft correspondence[END_REF] provides the convergence whereas the study of this solution shows that it expands as a generating series in the enumeration of some planar maps. Next orders can be derived by arguments similar to those developed for the GUE.

Approximate transport maps and universality

In this section, we discuss the generalization of transport maps to models which are approximately driven by the free difference quotient and show that this entails universality properties for the fluctuations of the eigenvalues.

β-models

Let us again consider the β-models given by

and recall that there is an equilibrium measure µ V such that lim

Theorem 13.

[3] Assume V,W are C 31 and satisfy Assumption 11. Then there exists

where . TV is the total variation norm. Moreover, there exists a positive finite constant C such that sup 1≤k≤N T N,k

In this theorem T 0 is simply the optimal transport map from the equilibrium measures µ V to µ W ; T 0 µ V = µ W . In particular T 0 is increasing. The proof of such a result follows the approach by Poisson equation developed in section 3.2. The idea is to write these equations for β-models and find an approximate solution by linearizing these equations around the limit, and showing that the leftovers only produce a small error in the density, hence producing a small error in total variation. Theorem 13 entails universality of the fluctuations, namely that the local fluctuations of the eigenvalues under P V N,β and P W N,β are the same. Taking V = 1 2 x 2 , it yields for instance the following theorem: Theorem 14. Let W be C 31 and satisfying Assumption 11. Assume that the eigenvalues are ordered λ 1 ≤ • • • ≤ λ N . Then, 1. if the support of µ V is given by [a, b], there is a constant ρ so that N 2/3 ρ(λ Na) converges in distribution to the Tracy-Widom law TW β . A similar result holds at the left edge b.

2. Take ε > 0 and i ∈ [εN, (1ε)N] so that i/N E. Then, there exists a constant c E so that Nc E (λ iλ i-1 ) fluctuates as in the case where W = x 2 2 , in particular following the Gaudin distribution if β = 2. Such a result was also proved in [START_REF] Bourgade | Edge universality of beta ensembles[END_REF][START_REF] Bourgade | Universality of general β-ensembles[END_REF] under weaker assumptions on V (but for β ≥ 1) and by Scherbina [START_REF] Shcherbina | Change of variables as a method to study general beta-models: bulk universality[END_REF] (in the bulk but including cases where the support of µ V is not connected, an extension that was also considered in [START_REF] Bekerman | Transport maps for β-matrix models in the multi-cut matrix regime[END_REF] by using approximate transport maps).

It is quite clear that approximate transport maps entail universality, as in the case where T N 1 is null, the eigenvalues λ W i under P W,β N are just push forward of those under P x 2 N so that

and a similar result holds inside the bulk. ( 34) allows indeed to neglect T N 1 .

Matrix models

One would like to understand as well the fluctuations of the eigenvalues of polynomials in several independent matrices from the GUE or of matrices randomly chosen according to [START_REF] Guionnet | Large random matrices: lectures on macroscopic asymptotics[END_REF]. Let us consider d independent matrices X N 1 , . . . , X N d taken from the GUE and P a polynomial in d non-commutative indeterminates. Then, it was shown (in fact also based on the analysis of Schwinger-Dyson equation) by Haagerup and Thorbjørnsen [START_REF] Haagerup | A new application of random matrices: Ext(C * red (F 2 )) is not a group[END_REF] that the operator norm of P(X N 1 , . . . , X N d ) converges almost surely towards its free limit:

One could wonder about the fluctuations around this limit. Of course, it should depend on properties of the polynomial P since even in the case d = 1, the largest eigenvalue of P(X N 1 ) could either be the image of the largest eigenvalue of X N 1 or of an eigenvalue inside the bulk. It would be interesting to understand what are the optimal conditions so that the fluctuations are still given by the Tracy-Widom law. With Figalli, we could prove that this is the case when P is a small perturbation of the identity.

Theorem 15. Let P be a self-adjoint polynomial. For ε small enough (independent of N), the eigenvalues of Y N = X N 1 + εP(X N 1 , . . . , X N d ) fluctuates locally as when ε = 0, that is following the sine-kernel in the bulk and the Tracy-Widom law at the edge.

In fact, for ε small enough, a change of variable shows that the law of Y N is of the type described in [START_REF] Guionnet | Large random matrices: lectures on macroscopic asymptotics[END_REF] (but with a slightly more complicated potential), hence we now focus on proving universality in the latter case. Take V a = d i=1 W i (X i ) + aW (X 1 , . . . , X d ) and consider

where integration holds over Hermitian or symmetric matrices. Denote P V a ,i N the law of the ordered eigenvalues of X N i under P V a N . The main result is that we can again construct approximate transport maps: Theorem 16 ( Figalli-G 2014). Assume V,W i smooth enough, W i ≥ c > 0.Then, there exists a(c) > 0 so that for a ∈ [-a(c), a(c)], there exist T 0 : R → R, T N 1 : R N → R N smooth functions so that

As a consequence, under P V a N , the law of the spacing distribution of X N i , Nc i j (λ i jλ i j+1 ), converges to the Gaudin distribution, and that of N 2/3 c i (max j λ i j -C i ) to the Tracy-Widom law, for some appropriate constants c i ,C i and c i j .

The idea to prove this theorem follows the arguments used in the previous section together with the fact that we can control integrals over the unitary groups by [START_REF] Guionnet | Asymptotics of unitary multimatrix models: The schwingerdyson lattice and topological recursion[END_REF], hence rewrite, the law of the eigenvalues P V a N of X 1 N , . . . , X d N under P N V a is )) exp{(N 2 F a 2 + NF a 1 + F a 0 )(

We are then in position to construct approximate transport maps as for β-models.


Pierre Prablanc
email: pierre.prablanc@technicolor.com

Alexey Ozerov
email: alexey.ozerov@technicolor.com

Ngoc Q K Duong

Patrick Pérez
email: patrick.perez@technicolor.com

TEXT-INFORMED SPEECH INPAINTING VIA VOICE CONVERSION

Keywords: Audio inpainting, speech inpainting, voice conversion, Gaussian mixture model, speech synthesis

The problem of speech inpainting consists in recovering some parts in a speech signal that are missing for some reasons. To our best knowledge none of the existing methods allows satisfactory inpainting of missing parts of large size such as one second and longer. In this work we address this challenging scenario. Since in the case of such long missing parts entire words can be lost, we assume that the full text uttered in the speech signal is known. This leads to a new concept of text-informed speech inpainting. To solve this problem we propose a method that is based on synthesizing the missing speech by a speech synthesizer, on modifying its vocal characteristics via a voice conversion method, and on filling in the missing part with the resulting converted speech sample. We carried subjective listening tests to compare the proposed approach with two baseline methods.

INTRODUCTION

The goal of audio inpainting consists in filling in missing portions of an audio signal. This concept was recently formulated by Adler et al. [START_REF] Adler | Audio inpainting[END_REF] as a general framework covering several existing audio processing problems such as audio declipping [START_REF] Abel | Restoring a clipped signal[END_REF], clicks removal [START_REF] Godsill | A Bayesian approach to the restoration of degraded audio signals[END_REF] and bandwidth extension [START_REF] Smaragdis | Missing data imputation for spectral audio signals[END_REF]. The term inpainting is borrowed from image inpainting [START_REF] Bertalmio | Image inpainting[END_REF], a similar problem in image processing, where the goal is to fill in missing parts in an image. The difficulty of an audio inpainting problem depends mainly on the nature of the signal (e.g., speech or music) and on the distribution of the missing parts (e.g., tiny holes of few samples or bigger holes of several milliseconds). For example IP packet losses in VoIP systems usually lead to missing intervals in the transmitted speech of length ranging from 5 ms to 60 ms. This problem is often addressed using packet loss concealment (PLC) algorithms [START_REF] Perkins | A survey of packet loss recovery techniques for streaming audio[END_REF][START_REF] Guoqiang | Autoregressive model-based speech packet-loss concealment[END_REF] that are only able to fill in the missing part with a quasi stationary signal. This is achieved either by repeating the last packet received [START_REF] Perkins | A survey of packet loss recovery techniques for streaming audio[END_REF] or by more sophisticated autoregressive model-based prediction/interpolation [START_REF] Guoqiang | Autoregressive model-based speech packet-loss concealment[END_REF]. A more advanced method consisting in smoothly filling-in the missing part with previously seen speech examples was recently proposed by Bahat et al. [START_REF] Bahat | Self-content-based audio inpainting[END_REF]. This method allows producing a more non-stationary and more natural signal in the missing part.

In this paper we address the problem of speech inpainting when the duration of the missing part may be very large (i.e., one or several seconds). Existing approaches such as PLC algorithms [START_REF] Perkins | A survey of packet loss recovery techniques for streaming audio[END_REF][START_REF] Guoqiang | Autoregressive model-based speech packet-loss concealment[END_REF] or example-based speech inpanting [START_REF] Bahat | Self-content-based audio inpainting[END_REF] are not designed to handle such This work was partially supported by ANR JCJC program MAD (ANR-14-CE27-0002). long missing areas. Indeed, whole words or big portion of a word may be entirely missing, and it often becomes not even clear what was really said. For example in "I ... you." sentence, the missing word (represented by dots) can be love, miss, hate, etc. To make the problem slightly better defined we assume that the text that should be pronounced in the missing part is known. As such, we assume that the text of the whole sentence is available (the text of the observed part may be always transcribed if needed). This leads to a so-called informed audio inpainting setup, where by analogy with informed or guided audio source separation [START_REF] Vincent | From blind to guided audio source separation[END_REF] some information about the missing signal is assumed known. More specifically, this particular audio inpainting setup is very close in spirit to text-informed audio source separation [START_REF] Magoarou | Textinformed audio source separation. Example-based approach using non-negative matrix partial co-factorization[END_REF].

A successful text-informed speech inpainting algorithm might be still applied for speech restoration in VoIP transmission, though it is not the most straightforward application. Indeed, first, the approach must operate online in this case and, second, the text needs to be known on the receiver side. However, there are several new applications that become possible. First, this new inpainting strategy may be used in the audio post-production workflows. One important and demanding task in audio post-production is the postsynchronization (a.k.a. additional dialogue recording) where actors must record again their lines in a studio because on-set recordings contain slight text errors or unexpected noise, or because dialogue changes are made a posteriori. Such problems could be partially addressed by the proposed technique instead. Similarly, dubbing often requires final edits to just slightly correct small portions of the new speech, a task that could benefit from our technique. Second, it would allow restoring beep censored speech in TV shows or movies by either reproducing the original or a modified speech in the bleeped part. Finally, text-informed speech inpainting could be suitable for various other speech editing needs, including the partial "rewriting" of speech sequences in the more general context of audio-visual content editing, e.g. [START_REF] Bregler | Video rewrite: Driving visual speech with audio[END_REF].

We propose a solution for text-informed speech inpainting that is based on speech synthesis [START_REF] Bastiaan | Speech coding and synthesis[END_REF] and voice conversion [START_REF] Stylianou | Continuous probabilistic transform for voice conversion[END_REF]. More precisely our approach is based on the following main steps:

1. A speech sample (source speech) corresponding to both observed and missing parts is synthesized given the text;

2. A voice conversion mapping is learned from observed parts of the speech to inpaint (target speech) and the corresponding parts of the synthesized speech;

3. The resulting voice conversion mapping is applied to the source speech parts corresponding to the missing parts of the targets speech;

4. The missing parts in the target speech are filled in with the obtained converted speech.

Another option we consider is when the source speech is not synthesized, but more naturally pronounced by a user. This may be possible within a user-assisted speech processing tool. The proposed method is somehow related to an analysis/synthesis-based speech enhancement method [START_REF] Carmona | Speech spectral envelope enhancement by HMM-based analysis/resynthesis[END_REF], though the problem considered in this latter work is speech enhancement, which is very different from speech inpainting considered here. The rest of the paper is organized as follows. Section 2 is devoted to a description of voice conversion in general and of the particular voice conversion method we used. Understanding voice conversion is necessary to further understand some particularities of the proposed speech inpainting method described in Section 3. Subjective listening tests were carried out to compare the proposed approach with two baselines: the source speech and the converted source speech. The results of subjective tests are presented in Section 4 and some conclusions are drawn in Section 5.

VOICE CONVERSION

In this section we recall main principles of voice conversion and describe a particular voice conversion system we used in this work.

Generalities

The general goal of voice conversion is to modify some characteristics of a speech signal such as speaker identity, gender, mood, age, accent, etc. [START_REF] Helander | Mapping Techniques for Voice Conversion[END_REF], while keeping unchanged the other characteristics including the linguistic information. In this work we are interested in speaker identity transfer. As such, the goal of voice conversion we consider here is to modify a speech signal uttered by a so-called source speaker so that it sounds as if it was pronounced by another so-called target speaker.

Most of voice conversion systems consist of the following two ingredients:

• Analysis / synthesis: An analysis system transforms the speech waveform into some other representation that is related to the speech production model, and thus easier to modify some speech characteristics. The corresponding synthesis system resynthesizes back a speech signal waveform from the transformed representation.

• Voice conversion mapping is applied to the transformed speech representation so as to modify its characteristics. This mapping is usually learned from some training data.

Analysis and synthesis

As analysis system we use the STRAIGHT-analysis [START_REF] Kawahara | Speech representation and transformation using adaptive interpolation of weighted spectrum: vocoder revisited[END_REF] that allows estimating the fundamental frequency f 0 and a smooth spectrum. We then compute the Mel-frequency cepstral coefficients (MFCCs) [START_REF] Vergin | Generalized mel frequency cepstral coefficients for large-vocabulary speakerindependent continuous-speech recognition[END_REF] from the STRAIGHT-spectrum. As such our transformed representation consists of f 0 and MFCCs. For synthesis we reconstruct STRAIGHT-spectrum from the MFCCs and then re-synthesize the speech waveform from STRAIGHT-spectrum and f 0 using STRAIGHT-synthesis [START_REF] Kawahara | Speech representation and transformation using adaptive interpolation of weighted spectrum: vocoder revisited[END_REF].

Voice conversion mapping

Many approaches were proposed to build voice conversion mapping including those based on Gaussian mixture models (GMMs) [START_REF] Stylianou | Continuous probabilistic transform for voice conversion[END_REF][START_REF] Toda | Voice conversion based on maximum-likelihood estimation of spectral parameter trajectory[END_REF], nonnegative matrix factorization (NMF) [START_REF] Aihara | A preliminary demonstration of exemplar-based voice conversion for articulation disorders using an individuality-preserving dictionary[END_REF], artificial neural networks (ANN) [START_REF] Narendranath | Transformation of formants for voice conversion using artificial neural networks[END_REF] and partial least squares regression [START_REF] Helander | Voice conversion using partial least squares regression[END_REF]. Here we have chosen to follow one of the most popular GMM-based approach proposed by Toda et al. [START_REF] Toda | Voice conversion based on maximum-likelihood estimation of spectral parameter trajectory[END_REF].

Modeling

As feature vector to be predicted we consider, as in [START_REF] Toda | Voice conversion based on maximum-likelihood estimation of spectral parameter trajectory[END_REF], the MFCCs (static features) concatenated with their derivatives (dynamic features). We use a so-called joint GMM modeling [START_REF] Helander | Mapping Techniques for Voice Conversion[END_REF] that models a joint distribution of source and target features [START_REF] Kain | Spectral voice conversion for text-tospeech synthesis[END_REF]. Moreover, we consider GMM with "tri-diagonal" covariance matrices, which are much more efficient to compute as compared to GMM with full covariance matrices.

Training

Voice conversion mapping is usually trained from a so-called parallel dataset, i.e., a set of sentences uttered by both source and target speakers. These sentences are first aligned using, e.g., the dynamic time warping (DTW) [START_REF] Müller | Dynamic time warping[END_REF] applied to the MFCCs. Then, a joint GMM is trained from the set of aligned and concatenated together source and target feature vectors using the expectation-maximization (EM) algorithm [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF].

Conversion

Once the joint GMM is learned, to convert a new source speech, target speech features are first predicted in the minimum mean square error (MMSE) sense, given the source features and the model [START_REF] Toda | Voice conversion based on maximum-likelihood estimation of spectral parameter trajectory[END_REF]. Finally, since most likely the predicted MFCCs and their derivatives do not correspond to any original MFCC sequence, the MFCCs are re-estimated in a maximum likelihood sense, thus introducing some temporal smoothness in their trajectories thanks to the derivatives [START_REF] Toda | Voice conversion based on maximum-likelihood estimation of spectral parameter trajectory[END_REF]. The f 0 is often not predicted by the GMM, but via a simpler linear regression in the logarithmic domain, and we follow the same strategy here.

PROPOSED SPEECH INPAINTING APPROACH

In this section a description of the proposed approach is given throughout the subsections below. First, it is assumed that a speech signal with a missing part is given and the exact location of this part is indicated. It is also assumed that the speech is pronounced by just one speaker. Following voice conversion terminology the speech signal is called target speech and the corresponding speaker is called target speaker. For the sake of simplicity the description below is given for the case when there is only one missing segment in the target speech signal.

Source speech sample production

Given the uttered text for both observed and missing parts, a source speech is produced, inline with [START_REF] Magoarou | Textinformed audio source separation. Example-based approach using non-negative matrix partial co-factorization[END_REF], either using a speech synthesizer or by a human operator within a user-guided tool. Both strategies have their pros and cons as follows. Within an application where a fully-automated process is needed the synthesis-based strategy is preferable, since it does not require any human intervention. However, the user-guided strategy may provide a source speech of a much higher quality. First, it is not synthetic. Second, in contrast to the synthesis-based approach, it can be much better adapted to the target speech rate, emotion, and other characteristics.

Alignment between source and target speech

Source and target speech signals are temporally aligned for the following two reasons:

• First, this alignment is needed to identify which portion of the source signal corresponds to the missing part of the target signal. This portion will be then used for inpainting.

• Second, these aligned signals are then used as a parallel dataset for training voice conversion mapping.

Note that this alignment is not trivial, since there is a missing part in the target speech and we also need to identify the corresponding part in the source speech. To achieve that we propose a simple strategy based on DTW that is visualized on Figure 1 and briefly described as follows. The missing part is simply removed from the target speech, but its location is retained. MFCCs are extracted from both source and target signals and a DTW path based on the distances between MFCC vectors is computed. One might expect that this path should form approximately a straight line (horizontal on Fig. 1) around the region corresponding to the missing part. As such, by introducing a small forward/backward tolerance around the known missing part position within the target signal, one can identify the corresponding source speech part as shown in the figure.

Voice conversion mapping training

Voice conversion mapping, in our case a GMM, is trained from the parallel data, i.e., the aligned source and target speech signals in the regions where the target signal is observed. Note that if some auxiliary target speech data with the corresponding text (transcription) is available, the training parallel dataset may be augmented to improve voice conversion performance.

Voice conversion

A source speech segment to be converted is extracted as follows. This segment must include the region corresponding to the missing part in the target speech (as identified in Section 3.2), but also must contain some signal before and after this region. This precaution of taking a slightly bigger segment is needed to assure a smooth transition during the inpainting. The extracted source speech segment is transformed by the analysis system. The resulting transformed representation, in our case MFCCs and f 0, is then converted using the pre-trained voice conversion mapping to make its characteristics closer to those of the target speaker.

Speech inpainting in the transformed domain

A transformed representation of the observed target speech parts is computed in its turn by the analysis system. In the transformed representation the converted source speech segment is inserted to fill in the missing part in the target speech in a smooth way via a fade-in fade-out interpolation strategy. This is possible thanks to the fact that the converted source speech segment is bigger than the missing region. Note though that, as for the spectral envelop parameters, we interpolate by fade-in fade-out directly the STRAIGHT-spectrum rather than the MFCCs. The resulting target speech waveform is resynthesized from the obtained inpainted transformed representation via the synthesis system.

EXPERIMENTS

Since to our best knowledge none of the existing speech inpainting approaches allows handling such long missing segments, we resort to two baselines for comparisons. Namely, we compare the following three methods:

• "Inpainted": the proposed approach;

• "Converted": the entire source speech (not only the missing part) that was converted by voice conversion;

• "Source": the non-processed source speech that was either synthesized or pronounced.

We compare these three approaches using a subjective listening test.

Data

We have created a dataset including natural speech samples of 4 English speakers (2 male and 2 female speakers) from the CMU ARCTIC database [START_REF] Kominek | The CMU Arctic speech databases[END_REF] and synthetic speech samples of 2 English speakers (1 male and 1 female speaker) synthesized by the IVONA speech synthesizers. 1 One male and one female speaker from the CMU ARCTIC database are always considered as target speakers and two other speakers are always considered as source speakers.

Both synthetic IVONA speakers are considered as source speakers. All the signals are sampled at 16 kHz. Each speech inpainting setup is characterized by a pair of different speakers (source and target) and includes:

• Target speech to inpaint: one approximately 3 sec long target speech sample with a missing segment that is chosen randomly, while assuring that it is not toot close to the border;

• Source speech: one approximately 3 sec long source speech sample uttering the same text as the target;

• Parallel dataset for training: an auxiliary parallel dataset of these two speakers for voice conversion training that does not include the above 3 sec long test speech samples;

• Target speech examples: two 3 sec long speech samples uttered by the target speaker that are different from the test samples to inpaint and are needed to allow the listening test participants judging the speaker identity preservation as it will be explained below. We then created the 16 different speech inpainting setups consisting of all possible combinations of the following binary choices:

1. Source and target speakers either both male or both female, Finally, none of inpainting setup pairs shared the same test sentence.

Parameters and simulations

For DTW alignment (Sec. 3.2) we used only 12 first MFCCs computed directly from the corresponding signals. For MFCCs computed from STRAIGHT-spectrum (Sec. 2.2) we kept all 40 coefficients, which allows quite precise, yet not perfect, STRAIGHTspectrum reconstruction from MFCCs during the resynthesis step. The joint GMM (Sec. 2.3.1) included 32 Gaussian components.

For each speech inpainting setup we ran the proposed inpainting approach together with voice conversion applied to the whole source speech signal, thus obtaining three speech samples: inpainted, converted and source as described in the beginning of Section 4. It should be noted that when the proposed missing part identification strategy (see Fig. 1) failed in correctly identifying the missing part, it was re-adjusted by hand. This happened for 6 out of 16 sequences.

Listening test and results

A total of 15 persons (2 women and 13 men) participated in the listening test. All the listeners used headphones. For each of the 16 speech inpainting setups presented in a random order, each participant first listened to two target speech examples, to have an idea about the target speaker identity. The participant would then listen to the three speech samples (inpainted, converted and source) presented in a random order, and was asked to note for each sample both the speech audio quality ("Does it sounds natural? Can you hear artefacts or not?") and the identity preservation ("Does the voice in the samples resemble to the voice of the target speaker?") on a 0 to 5 scale (greater is better). The random orders of presentation were different for different participants. The participants were not informed about the nature of the processing of speech samples. We have decided not keeping the results of participants who have noted at least once a natural source speech quality smaller than 4 or smaller than the quality of a processed sample (inpainted or converted). After such a filtering we have retained the results of 12 participants. Note that in this experiments we did a deliberate choice not to compare the original speech having missing parts, since it is meaningless to perform such a comparison in terms of the speech quality and the identity preservation. Indeed, for example if the missing part corresponds to just one entire word in the sentence, removing this word from the speech sample would not affect neither the quality nor the identity preservation.

Figure 2 summarizes the test results averaged over 12 participants and over different conditions. As expected the source has always the best quality (Fig. 2 (a)). However, the inpainted speech has a better quality than the converted one. Moreover, the inpainted speech outperforms the two baselines in terms of identity preservation. One can note from Fig. 2 (b) that inpainting long missing segments leads to a quality degradation as compared to inpainting short missing segments. As for other conditions (Figs. 2 (c) and (d)), they do not seem to influence too much the results.

CONCLUSION

In this paper we have formulated the problem of text-informed speech inpainting, where it becomes potentially possible to perform a satisfactory inpainting of quite long missing parts (several seconds) in a speech signal thanks to the knowledge of the uttered text. This new framework opens a door for new speech editing capacities and can be applied, e.g., for post-sync and dubbing. We have proposed a solution for this problem based on voice conversion. Experimental results have shown that the proposed speech inpainting approach leads to both better speech quality and better speaker identity preservation as compared to using voice conversion alone. Further work will include research towards a complete automation of the inpainting process and the inpainted speech quality improvement. Another interesting research path would be to consider a similar problem in music processing: a score-informed music inpainting.

ACKNOWLEDGEMENT

The authors would like to thank colleagues from Technicolor who participated in the listening test.

Fig. 1 :

 1 Fig. 1: Source vs. target alignment and missing part identification.

1Fig. 2 :

 2 Fig. 2: Listening test results averaged over 12 test participants and over different conditions.

2 .

 2 Missing segment either long (750 ms) or short (250 ms); 3. Parallel training dataset either small (30 sec) or big (5 min); 4. Source speech either natural or synthetic.