Columbia Lectures notes I wrote lecture notes on the uses of Dyson-Schwinger equations after a course at Columbia University in august 2017.

Rigidity and universality at the edge for discrete Beta-ensembles With Jiaoyang Huang, we prove that discrete Beta-ensembles are rigid and deduce that fluctuations at the boundary of the liquid region are driven by the Tracy-Widom laws by comparison to the continuous case.

Discrete-Beta ensembles, With Vadim Gorin and Alexei Borodin, we use Nekrasov's equations to study the global fluctuations of discrete Beta-ensembles.

Transport maps for Beta-matrix models and Universality With Florent Bekerman and Alessio Figalli we construct approximate transport maps between Coulomb gas interacting particle systems. This implies universality of the fluctuations of the spacings and extreme eigenvalues.

Central limit theorems for linear statistics of heavy tailed random matrices, Central limit theorem for eigenvectors of heavy tailed matrices We study the central limit theorem for linear statistics of heavy tailed random matrices. A slight generalization allows to catch the marcoscopic fluctuations of their eigenvectors.

Asymptotic expansion of beta matrix models in the one -cut regime ,Asymptotic expansion of beta matrix models in the multi-cut regime,Large-N asymptotic expansion for mean field models with Coulomb gas interaction In this series of articles, we develop a technique to prove large $N$ expansion of mean field interacting particle systems with a Coulomb gas interaction. In the first we consider the case where the limiting measure has a connected support, in the second it can have several connected components in its support. In the last article we consider the case where the potential is not anymore a linear function of the empirical measure, hence allowing general smooth interactions between particles.

PRL EJP In these articles, we construct the Beta- Dyson Brownian motion from the case where beta is equal to two by tossing a coin at every small step of time independently to decide whether the evolution will follow a brownian motion or a Hermitian Brownian motion.

Free transport In this article, we adapt ideas from classical probability to construct a transport map between two non-commutative law in a perturbative regime. This in particular implies the isomorphisms of q- Gaussian variables when q is small enough.

Localization and delocalization of eigenvectors for heavy-tailed random matrices written with C. Bordenave considers the eigenvectors of matrices with i.i.d heavy tailed entries. We show some localization if the enrties have finite expectation and a weak form of delocalization otherwise.

Convergence of the spectral measure of non normal matrices studies the regularization of the spectral measure of non-normal matrices by Gaussian matrices.

Support convergence in the single ring theorem is a follow up of the article written with M. Krishnapur and O. Zeitouni where this time we prove convergence of the support. 2010.

Loop models, random matrices and planar algebras is an article written with V. Jones, D. Shlyakhtenko and P. Zinn Justin on the construction of traces on planar algebra including a potential that can be obtained as a limit of matrix models. This construction generalizes for instance the famous $O(n)$ models and the Potts model on random graphs. We discuss how to solve some of these models.2010.

Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices and Large deviations of the extreme eigenvalues of random deformations of matrices are two articles written in collaboration with F. Benaych Georges and M. Maida on the fluctuations and the large deviations of the extreme eigenvalues of a deterministic matrix by a finite rank perturbation. Quite a few results extend to the case where the full rank matrix is random and taken according to the classical Gaussian ensembles of matrices, 2010.

Wigner matrices is a survey written in collaboration with G. Ben Arous on Wigner matrices, 2010.

Proceedings de l'ICMP is the proceedings for ICMP, Prague 2009.

Bourbaki is the proceedings for Bourbaki summarizing the amazing breakthrough on universality by Erdos-Yau et al and Tao-Vu et al in 2009.

The single ring theorem with M. Krishnapur and O. Zeitouni (2009). We study the distribution of the (complex) eigenvalues of square non-normal random matrices of the form UDV with U,D following the Haar measure on the unitary group and show that asymptotically they are supported by a connected rotationnally invariant set. To appear in Ann. Math.

One can find on arxiv two papers about random matrices with heavy tailed entries written respectively with Gerard Ben Arous and Amir Dembo and Serban Belinschi, both published in Communications in mathematical physics (2008 and 2009 respectively). With Gerard ben Arous, we show that the spectral measure of Wigner matrices with heavy tailed entries converges towards a heavy tailed law, which puts on a firm mathematical ground an article by Cizeau and Bouchaud. With Amir Dembo and Serban Belinschi, we extend these results to band matrices and Wishart matrices.

Asymptotics of unitary and orthogonal matrix integrals is a paper written with B. Collins and E. Maurel Segala, published by Advances in Mathematics (2010), that generalizes my joint works with E. Maurel Segala to integrals over the unitary and orthogonal groups equipped with their Haar measure. We give also a combinatorial interpretation of these integrals that apply in particular to Harich-Chandra--Itzykson-Zuber integral.

Regularization by free additive convolution, square and rectangular cases .ps with Serban Belinschi and Florent Benaych-Georges(Complex Analysis and Operator theory (2009)) We study the effect of free convolution for the law of rectangular matrices, and in particular the existence, regularity and vanishing of the resulting density.

Free diffusions and Matrix models with strictly convex interaction .ps ( GAFA (2009)) with D. Shlyakhtenko

We study general limits of matrix models with a convex potential. One of the fun remark is that the Schwinger Dyson equation (which in the case of one matrix means that the Hilbert transform of a measure is given by a polynomial on the support of this measure) determines uniquely the tracial state when the polynomial is convex. From there, we deduce analyticity of the resulting states in terms of the coefficients of the polynomials, connectivity of the support of the matrices etc etc

On Classical Analogues of Free Entropy Dimension .ps ( JFA (2007) ) with D. Shlyakhtenko

We study the analogue of the entropy dimension introduced by D. Voiculescu in the classical framework. We show that it is then related with the grows of the volume of balls and deduce that it is invariant by Lipschitz maps. We propose diverse approaches to this quantity, one based on Bochner inequality.

Second order asymptotics for matrix models ( Annals of Probability (2007)) with E. Segala Maurel

We obtain the second order expansion in a general matrix model as well as a central limit theorem.

Combinatorial aspects of matrix models .ps (Alea (2006)) with E. Segala Maurel

We show that under reasonably general assumptions, the first order asymptotics of the free energy of matrix models are generating functions for colored planar maps. This is based on the fact that solutions of the Schwinger-Dyson equations are, by nature, generating functions for enumerating planar maps, a remark which bypasses the use of Gaussian calculus.

Timescales of population rarity and commonness in random environments with R. Ferriere and I. Kurkova

This paper investigates the influence of environmental noise on the characteristic timescale of the dynamics of density-dependent populations. General results are obtained on the statistics of time spent in rarity (i.e.\ below a small threshold on population density) and time spent in commonness (i.e. above a large threshold).

Cugliandolo-Kurchan equations for dynamics of Spin-Glasses with G. Ben Arous and A. Dembo (Prob. Th. and rel. Fields )(2005)

We study the Langevin dynamics for the family of spherical $p$-spin disordered mean-field models of statistical physics. We prove that in the limit of system size $N$ approaching infinity, the empirical state correlation and integrated response functions for these $N$-dimensional coupled diffusions converge almost surely and uniformly in time, to the non-random unique strong solution of a pair of explicit non-linear integro-differential equations, first introduced by Cugliandolo and Kurchan.

Large deviations and stochastic calculus for large random matrices (Probability surveys) (2004) . These are lecture notes of a course I gave in Brazil during summer 2003. They intend to present large deviations techniques for large random matrices quantities such as their spectral measure. They are supposed to be accessible to non-probabilists and non free-probabilists.

A probabilistic approach to some problems in von Neumann algebras Proceedings ECM4 (04).

Aging Proceedings ICIAM 02.

A Fourier view on the $R$-transform and related asymptotics of spherical integrals (Journal of Functionnal analysis)(2005) with M. Maida. We estimate the asymptotics of spherical integrals when the rank of one matrix is much smaller than its dimension. We show that it is given in terms of the $R$-transform of the spectral measure of the full rank matrix and give a new proof of the fact that the $R$-transform is additive under free convolution. These asymptotics also extend to the case where one matrix has rank one but complex eigenvalue. We are very grateful to Alexei Onatski for pointing out to us a typo in the published version of this paper in theorem 3.

Long time behaviour of the solution to non-linear Kraichnan equations , with C. Mazza (Probability Theory and related fields) (2005) . The dynamics of spherical SK p-spins models are described by a system of non linear integro-differential equations relating the dynamical covariance and the so-called response function. The long time behaviour of the solution, which should describe the aging of the system, has not yet been satisfyingly extracted from this system despite very interesting articles of L. Cugliandolo and J. Kurchan. In this article, we try to analyze the behaviour of the response-function when the asymptotic behaviour of the covariance is prescribed and relate this last question with the asymptotic behaviour of a non-commutative process.

Character expansion method for the first order asymptotics of a matrix integral , with M. Maida (Prob. Th. rel. Fields)(2004) . We study the first order asymptotic of a matrix model related with the dually weighted graph model considered by Kazakov, Staudacher and Wynter. It is based on character expansion as well as the control on infinite positive sums over Young tableaux weighted by Schur functions.

Addendum to: Large deviations asymptotics for spherical integrals , with O. Zeitouni (to Journal of functionnal analysis )(2004) . We improve our previous result by showing that the full large deviations principle do not only holds on the law of on time marginal but on the law of the whole measure valued-process.

First order asymptotics of matrix integrals ; a rigorous approach towards the understanding of matrix models , (Comm. Math. Phys., 244, 527-569) (2003) The original publication is available at http:www.springerlink.com We use the asymptotics of spherical integrals obtained with O. Zeitouni to study the asymptotics of matrix integrals with AB interaction. The main new input is the study of the minimizing path of the rate function which is shown to be unique and described by a complex Burgers equation.

Large deviation bounds for matrix Brownian motion with P. Biane and M. Capitaine (Inventiones Mathematicae, 152, 433-459) (2003) We show that the microstates entropy is bounded above by the so-called microstates-free entropy, and bounded below by another non trivial entropy. We use large deviation techniques. The main technical input is the idea to generalize hydrodynamics technics to empirical measures on path space by constructing exponential martingales based on Clark-Ocone formula.

Moderate Deviations for the Spectral Measure of certain Random Matrices, , with A. Dembo and O. Zeitouni ( Inst. H. Poincaré,39, 1013-1042 )(2002). We study the moderate deviations for the spectral measure of non-centered Gaussian Wigner matrices. The proof follows martingales techniques and the path representation of the law of the eigenvalues.

Large deviations asymptotics for spherical integrals , with O. Zeitouni ( Journal of functionnal analysis, 188, 461-515 (2001) ). We study the first order asymptotics of spherical integrals, called in physics Harich-Chandra or Itzykson-Zuber integrals. We prove some result announced by Matytsin in the unitary case and extend it to the orthogonal case.

Concentration of the spectral measure for large matrices with O. Zeitouni ( Electronic Communications in Probability (2000) ). We show that standard concentration techniques can be applied to study the concentration of the spectral measure or the normalized trace of words of random band matrices.

Aging of spherical spin glasses , with G. Ben Arous and A. Dembo (prob. th. rel. fields, 120, 1-67 (2001) ). We study the aging property of the simplest model of spherical SK model of spin glass. This proves completely an article of L. Cugliandolo and Dean. Despite the results are not hard to guess, the final proofs are rather technical.

Discussion around Voiculescu's free entropies , with T. Cabanal Duvillard (Advances in mathematics, 174,167-226) (2003). We prove some results around free entropy and free Fokker Planck equations. Despite these results, let us quote the fact that tracial states with non-commutative Hilbert transforms given by the cyclic derivative of a polynomial are dense in the set of tracial states with finite entropy and the description of the convolution of free Fokker Planck equation whose understanding is crucial to develop large deviations techniques.

Large deviations upper bounds for the laws of matrix-valued processes and non-commutative entropies, , with T. Cabanal Duvillard (Annals of Probability, vol 29, no. 3, 1205-1261 (2001) ). This is the first paper of the serie where the basis to study spherical integrals and microstates entropy were drawn. The paper with O. Zeitouni on spherical integrals was an improvment of part of the first part of this article whereas that with P. Biane and M. Capitaine improves the second part.

Large deviations upper bounds and central limit theorems for band matrices and non-commutative functionnals of Gaussian large random matrices .( Annales de l'Institut Henri Poincaré , vol 38, no. 3, 341-384(2002)) We apply the method introduced in the article with T. Cabanal Duvillard to study Gaussian band matrices.

Notes de cours concernant les inégalités de Log-Sobolev, with B. Zegarlinski ( séminaires des probabilités, LNM XXXVI).

Stability of precise Laplace's method under approximations ; Applications (Version corrigée)

Large Deviations for Interacting Particle Systems. Applications to Non Linear Filtering avec P. Del Moral (Version corrigée) Publié a Stoch. Processes and Applications, vol. 78, 69-95 (1998)

Large Deviations for Interacting Particle Systems.