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 work, we developp 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.
put nctransport.pdfIn 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.
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 independnetly to decide whether the evolution will
follow a brownian motion or a Hermitian Brownian motion.
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
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.
Grandes matrices aléatoires et théorèmes d'universalité
is the proceedings for an Exposé Bourbaki which took place march 13 2010.
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
Regularization by free additive convolution, square and rectangular
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 trcial 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
( 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
Timescales of population rarity and commonness in
random environments .ps
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.
are obtained on the statistics of time spent in rarity (i.e.\
a small threshold on
population density) and time spent in commonness (i.e. above a
Cugliandolo-Kurchan equations for dynamics of Spin-Glasses .ps
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
first introduced by Cugliandolo and Kurchan.
deviations and stochastic calculus for large random matrices .ps
(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
A probabilistic approach to some problems
in von Neumann algebras
Proceedings ECM4 (04).
Proceedings ICIAM 02.
A Fourier view on the $R$-transform and related
asymptotics of spherical integrals .pdf
(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.
behaviour of the solution to non-linear Kraichnan equations , .ps 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 , .ps
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.
to: Large deviations asymptotics for spherical integrals , .ps
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.
order asymptotics of matrix integrals ; a rigorous approach towards
the understanding of matrix models , .ps (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.
deviation bounds for matrix Brownian motion, .ps 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.
Deviations for the Spectral Measure of certain Random Matrices, , .ps,
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.
deviations asymptotics for spherical integrals , .ps
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
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.
spherical spin glasses , .ps ,
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
Discussion around Voiculescu's free entropies , .ps ,
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 upper bounds for the laws of matrix-valued processes and
non-commutative entropies, .ps, avec 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.
deviations upper bounds and central limit theorems for band matrices
and non-commutative functionnals of Gaussian large random matrices .ps (
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.
cours concernant les inégalités de Log-Sobolev, with B. Zegarlinski (
séminaires des probabilités, LNM XXXVI).
of precise Laplace's method under approximations ; Applications (Version
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)
Deviations for Interacting Particle Systems.