Publications

You can find most of my recent articles on the arXiv and HAL

• Spectral Phase Transitions in Non-Linear Wigner Spiked Models, with Justin Ko, Florent Krzakala, Pierre Mergny and Lenka Zdeborová. We study the BBP transition for a matrix obtained by applying a non-linearity to the entries of a Wigner matrix with a rank one perturbation
• Estimating rank-one matrices with mismatched prior and noise: universality and large deviations, , Low-rank Matrix Estimation with Inhomogeneous Noise, with Justin Ko, Florent Krzakala and Lenka Zdeborová. We study low rank matrix estimation in the case of inhomogeneous noise and when prior and noise are mismatched.
• Full large deviation principles for the largest eigenvalue of sub-Gaussian Wigner matrices, with Nick Cook and Raphael Ducatez. We complete the large deviations principle for the largest eigenvalue of subGaussian Wigner matrices in great generality
• Matrix Models at low temperature, with Edouard Maurel Segala. We study the convergence of the empirical measure of the eigenvalues ofseveral matrix models in non-convex situations where a parameter goes to infinity.
• Large Deviations Principles via Spherical Integrals, with Serban Belinschi and Jiaoyang Huang (PMP 2022) We use the asymptotics of spherical integrals to derive large deviations for the empirical measure of the diagonal entries of a randomly rotated matrix, and large deviations estimates for the emprical measure of the eigenvalues of A+UBU*. We extend these ideas to estimate the asymptotics of Kotska numbers and Littlewood-Richardson coefficients
• Large Deviations for the largest eigenvalue of Sub-Gaussian Matrices, with Fanny Augeri and Jonathan Husson (CMP 2021) We derive large deviations estimates for the largest eigenvalue of Wigner matrices with sub-Gaussian entries by tilting the measure by spherical integrals: we show it differs from the Gaussian one at list for large enough deviation in the general case.
• Large deviations for the largest eigenvalue of Rademacher matrices, with Jonathan Husson (AoP 2020) We show that if the entries of a Wigner matrix have entries whose Laplace transform are bounded by that of the Gaussian with the same variance, then the largest eigenvalue obeys a large deviation principle with the same rate function than in the Gaussian case.
• Large deviations for the largest eigenvalues and eigenvectors of spiked Gaussian random matrices, with Giulio Biroli We study the joint large deviations for the first time.
• Large deviations for the largest eigenvalue of the sum of two random matrices, with Mylene Maida We study the large deviations for the largest eigenvalue of A+UBU* by tilting via spherical integrals.
• On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices. This article contains a new proof of Haagerup and Thorbjornsen's result of strong convergence of a family of independent GUE matrices, that uses neither the linearization trick, nor the Stieljes transform. Our new approach allows to obtain new bounds for independent matrices and tensor matrices, including in new regimes.
• Columbia Lectures notes I wrote lectu\re notes on the uses of Dyson-Schwinger equations after a course at Columb\ia University in august 2017.
• Large deviations for generalized Gibbs ensembles of the classical Toda chain with Ronan Memin We prove and generalize a result by Herbert Spohn on the Generalized Gibbs ensembles for Toda chain by using large deviations for beta ensembles.
• Rigidity and\ universality at the edge for discrete Beta-ensembles With Jiaoyang Hu\ang, we prove that discrete Beta-ensembles are rigid and deduce that fluct\uations 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 Borodi\n, we  use Nekrasov's equations to study the global fluctuations of d\iscrete Beta-ensembles.
• Transport maps for Beta-matrix models and Universality With Florent Bekerman and Alessio Figalli we construct approximate transpo\rt maps between Coulomb gas interacting particle systems. This implies uni\versality 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 tailedrandom 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.
• PRLEJPIn 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 willfollow a brownian motion or a Hermitian Brownian motion.
• Free transportIn this article, we adapt ideas from classical probability to construct a transport mapbetween 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 Gaussianmatrices.
• 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 ofthe extreme eigenvalues of random deformations of matrices are two articles written in collaborationwith 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 collaborationwith 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 theoremwith M. Krishnapur and O. Zeitouni (2009). We study the distribution of the (complex) eigenvalues of square non-normal random matrices of theform UDV with U,D following the Haar measure on the unitary groupand 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 Arousand 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 entriesconverges 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 toHarich-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 theeffect of free convolution for the law of rectangular matrices,and in particular the existence, regularity andvanishing 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 thatthe Schwinger Dyson equation (which in the case of one matrixmeans 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 supportof the matrices etc etc
• On Classical Analogues of Free Entropy Dimension .ps ( JFA (2007) )with D. Shlyakhtenko
• We study the analogue of the entropy dimensionintroduced by D. Voiculescu in the classical framework.We show that it is then related with the grows of the volume of balls and deduce thatit is invariant by Lipschitz maps. We propose diverse approaches to thisquantity, 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 generalmatrix 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 generalassumptions, the first order asymptotics of the free energy of matrix models are generating functions for colored planarmaps. This is based on the fact thatsolutions of the Schwinger-Dyson equationsare, by nature, generating functionsfor enumerating planar maps, a remark which bypassesthe use of Gaussian calculus.
• Timescales of population rarity and commonness in random environments with R. Ferriere and I. Kurkova
• This paper investigates theinfluence 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 onpopulation 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. andrel. Fields )(2005)
• We study the Langevin dynamics for the family ofspherical $p$-spin disordered mean-field models of statistical physics. We prove that in the limit of system size $N$ approaching infinity, the empirical state correlation andintegrated response functions for these $N$-dimensional coupled diffusions convergealmost 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.
• Largedeviations and stochastic calculus for large random matrices (Probability surveys)(2004) . These are lecture notes of a course I gave in Brazil duringsummer 2003. They intend to present large deviations techniques forlarge random matrices quantities such as their spectral measure. Theyare supposed to be accessible to non-probabilists and nonfree-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 relatedasymptotics of spherical integrals (Journal of Functionnal analysis)(2005)with M. Maida. We estimate the asymptotics of sphericalintegrals when the rank of one matrixis much smaller than its dimension. We show that it is given interms of the $R$-transform of the spectralmeasure of the full rank matrixand give a new proof of thefact that the $R$-transform is additiveunder free convolution. These asymptotics also extend to the case whereone matrix has rank one but complex eigenvalue. We are very grateful to Alexei Onatski for pointing out to us a typo in the published versionof this paper in theorem 3.
• Long timebehaviour of the solution to non-linear Kraichnan equations , with C.Mazza (Probability Theory and related fields) (2005) . The dynamics of spherical SK p-spins modelsare described by a system of non linear integro-differential equationsrelating the dynamical covariance and the so-called response function.The long time behaviour of the solution, which should describe the agingof the system, has not yet been satisfyingly extracted from this systemdespite very interesting articles of L. Cugliandolo and J. Kurchan. In this article, we try to analyze the behaviour of theresponse-function when the asymptotic behaviour of the covariance isprescribed and relate this last question with the asymptotic behaviourof a non-commutative process.
• Character expansion method for the first order asymptotics of a matrixintegral , with M. Maida (Prob. Th. rel. Fields)(2004) . We study the first orderasymptotic of a matrix model related with the dually weighted graphmodel considered by Kazakov, Staudacher and Wynter. It is based oncharacter expansion as well as the control on infinite positive sumsover Young tableaux weighted by Schur functions.
• Addendumto: Large deviations asymptotics for spherical integrals ,with O. Zeitouni (toJournal of functionnal analysis )(2004) . We improve ourprevious result by showing that the full large deviations principle donot only holds on the law of on time marginal but on the law of thewhole measure valued-process.
• Firstorder asymptotics of matrix integrals ; a rigorous approach towardsthe understanding of matrix models , (Comm.Math. Phys., 244, 527-569) (2003) The original publication is availableat http:www.springerlink.com We use the asymptotics of sphericalintegrals obtained with O. Zeitouni to study the asymptotics ofmatrix integrals with AB interaction. The main new input is the studyof the minimizing path of the rate function which is shown to be uniqueand described by a complex Burgers equation.
• Largedeviation bounds for matrix Brownian motion with P. Biane and M. Capitaine (Inventiones Mathematicae, 152, 433-459) (2003) We show that themicrostates entropy is bounded above by the so-called microstates-freeentropy, and bounded below by another non trivial entropy. We use largedeviation techniques. The main technical input is the idea togeneralize hydrodynamics technics to empirical measures on path spaceby constructing exponential martingales based on Clark-Ocone formula.
• ModerateDeviations for the Spectral Measure of certain Random Matrices, ,with A. Dembo and O. Zeitouni (Inst. H. Poincaré,39, 1013-1042 )(2002). We study the moderatedeviations for the spectral measure of non-centered Gaussian Wignermatrices. The proof follows martingales techniques and the pathrepresentation of the law of the eigenvalues.
• Largedeviations asymptotics for spherical integrals ,with O. Zeitouni (Journal of functionnal analysis, 188, 461-515 (2001) ). We study thefirst order asymptotics of spherical integrals, called in physicsHarich-Chandra or Itzykson-Zuber integrals. We prove some resultannounced by Matytsin in the unitary case and extend it to theorthogonal case.
• Concentration of thespectral measure for large matrices with O. Zeitouni (Electronic Communications in Probability (2000) ). We show thatstandard concentration techniques can be applied to study theconcentration of the spectral measure or the normalized trace of wordsof random band matrices.
• Aging of spherical spin glasses , with G. BenArous and A.Dembo (prob. th. rel. fields, 120, 1-67 (2001) ). We study theaging property of the simplest model of spherical SK model of spinglass. This proves completely an article of L. Cugliandolo and Dean.Despite the results are not hard to guess, the final proofs are rathertechnical.
• 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 Planckequations. Despite these results, let us quote the fact that tracialstates with non-commutative Hilbert transforms given by the cyclicderivative of a polynomial are dense in the set of tracial states withfinite entropy and the description of the convolution of free FokkerPlanck equation whose understanding is crucial to develop largedeviations techniques.
• Largedeviations upper bounds for the laws of matrix-valued processes andnon-commutative entropies, , with T. CabanalDuvillard (Annals of Probability, vol 29, no. 3, 1205-1261 (2001)). This is the first paper of the serie where the basis to studyspherical integrals and microstates entropy were drawn. The paper withO. Zeitouni on spherical integrals was an improvment of part of thefirst part of this article whereas that with P. Biane and M. Capitaineimproves the second part.
• Largedeviations upper bounds and central limit theorems for band matricesand 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 decours concernant les inégalités de Log-Sobolev, with B. Zegarlinski ( séminaires des probabilités, LNM XXXVI).
• Stabilityof precise Laplace's method under approximations ; Applications (Versioncorrigée)
• LargeDeviations for Interacting Particle Systems. Applications to NonLinear Filtering avec P. Del Moral (Versioncorrigée) Publié a Stoch. Processes and Applications,vol. 78, 69-95 (1998)
• LargeDeviations for Interacting Particle Systems.