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Alice Guionnet

Publications and source records attributed to Alice Guionnet.

At least 55 records · Page 3Linked to original sources
arXiv2014

Asymptotics of unitary multimatrix models: The Schwinger-Dyson lattice and topological recursion

We prove the existence of a 1/N expansion in unitary multimatrix models which are Gibbs perturbations of the Haar measure, and express the expansion coefficients recursively in terms of the unique solution of a noncommutative initial value problem. The recursion obtained is closely related to the "topological recursion" which underlies the asymptotics of many random matrix ensembles and appears in diverse enumerative geometry problems, but has not previously appeared in the context of random unitary matrices. Our approach consists of two main ingredients: an asymptotic study of the Schwinger-Dyson lattice over noncommutative Laurent polynomials, and uniform control on the cumulants of Gibbs measures on product unitary groups. The required cumulant bounds are obtained by concentration of measure arguments.

math-ph↗
arXiv2013

Central limit theorems for linear statistics of heavy tailed random matrices

We show central limit theorems (CLT) for the Stieltjes transforms or more general analytic functions of symmetric matrices with independent heavy tailed entries, including entries in the domain of attraction of $α$-stable laws and entries with moments exploding with the dimension, as in the adjacency matrices of Erdös-Rényi graphs. For the second model, we also prove a central limit theorem of the moments of its empirical eigenvalues distribution. The limit laws are Gaussian, but unlike to the case of standard Wigner matrices, the normalization is the one of the classical CLT for independent random variables.

math.PR↗
arXiv2012

Asymptotic expansion of beta matrix models in the one-cut regime

We prove the existence of a 1/N expansion to all orders in beta matrix models with a confining, off-critical potential corresponding to an equilibrium measure with a connected support. Thus, the coefficients of the expansion can be obtained recursively by the "topological recursion" of Chekhov and Eynard. Our method relies on the combination of a priori bounds on the correlators and the study of Schwinger-Dyson equations, thanks to the uses of classical complex analysis techniques. These a priori bounds can be derived following Boutet de Monvel, Pastur and Shcherbina, or for strictly convex potentials by using concentration of measure. Doing so, we extend the strategy of Guionnet and Maurel-Segala, from the hermitian models (beta = 2) and perturbative potentials, to general beta models. The existence of the first correction in 1/N has been considered previously by Johansson and more recently by Kriecherbauer and Shcherbina. Here, by taking similar hypotheses, we extend the result to all orders in 1/N.

math.PR↗
arXiv2012

Invariant $β$-ensembles and the Gauss-Wigner crossover

We define a new diffusive matrix model converging towards the $β$ -Dyson Brownian motion for all $β\in [0,2]$ that provides an explicit construction of $β$-ensembles of random matrices that is invariant under the orthogonal/unitary group. For small values of $β$, our process allows one to interpolate smoothly between the Gaussian distribution and the Wigner semi-circle. The interpolating limit distributions form a one parameter family that can be explicitly computed. This also allows us to compute the finite-size corrections to the semi-circle.

math.PR↗
arXiv2012

A diffusive matrix model for invariant $β$-ensembles

We define a new diffusive matrix model converging towards the $β$-Dyson Brownian motion for all $β\in [0,2]$ that provides an explicit construction of $β$-ensembles of random matrices that is invariant under the orthogonal/unitary group. We also describe the eigenvector dynamics of the limiting matrix process; we show that when $β< 1$ and that two eigenvalues collide, the eigenvectors of these two colliding eigenvalues fluctuate very fast and take the uniform measure on the orthocomplement of the eigenvectors of the remaining eigenvalues.

math.PR↗
arXiv2012

Localization and delocalization of eigenvectors for heavy-tailed random matrices

Consider an n x n Hermitian random matrix with, above the diagonal, independent entries with alpha-stable symmetric distribution and 0 < alpha < 2. We establish new bounds on the rate of convergence of the empirical spectral distribution of this random matrix as n goes to infinity. When 1 < alpha < 2 we give vanishing bounds on the Lp-norm of the eigenvectors normalized to have unit L2-norm goes to 0. On the contrary, when 0 < alpha < 2/3, we prove that these eigenvectors are localized.

math.PR↗
arXiv2011

Convergence of the spectral measure of non normal matrices

We discuss regularization by noise of the spectrum of large random non-Normal matrices. Under suitable conditions, we show that the regularization of a sequence of matrices that converges in *-moments to a regular element $a$, by the addition of a polynomially vanishing Gaussian Ginibre matrix, forces the empirical measure of eigenvalues to converge to the Brown measure of $a$.

math.PR↗
arXiv2011

Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices

Consider a deterministic self-adjoint matrix X_n with spectral measure converging to a compactly supported probability measure, the largest and smallest eigenvalues converging to the edges of the limiting measure. We perturb this matrix by adding a random finite rank matrix with delocalized eigenvectors and study the extreme eigenvalues of the deformed model. We give necessary conditions on the deterministic matrix X_n so that the eigenvalues converging out of the bulk exhibit Gaussian fluctuations, whereas the eigenvalues sticking to the edges are very close to the eigenvalues of the non-perturbed model and fluctuate in the same scale. We generalize these results to the case when X_n is random and get similar behavior when we deform some classical models such as Wigner or Wishart matrices with rather general entries or the so-called matrix models.

math.PR↗
arXiv2011

Large deviations of the extreme eigenvalues of random deformations of matrices

Consider a real diagonal deterministic matrix $X_n$ of size $n$ with spectral measure converging to a compactly supported probability measure. We perturb this matrix by adding a random finite rank matrix, with delocalized eigenvectors. We show that the joint law of the extreme eigenvalues of the perturbed model satisfies a large deviation principle in the scale $n$, with a good rate function given by a variational formula. We tackle both cases when the extreme eigenvalues of $X_n$ converge to the edges of the support of the limiting measure and when we allow some eigenvalues of $X_n$, that we call outliers, to converge out of the bulk. We can also generalise our results to the case when $X_n$ is random, with law proportional to $e^{- n Trace V(X)}\ud X,$ for $V$ growing fast enough at infinity and any perturbation of finite rank.

math.PR↗
arXiv2010

Support convergence in the single ring theorem

We study the eigenvalues of non-normal square matrices of the form A_n=U_nT_nV_n with U_n,V_n independent Haar distributed on the unitary group and T_n real diagonal. We show that when the empirical measure of the eigenvalues of T_n converges, and T_n satisfies some technical conditions, all these eigenvalues lie in a single ring.

math.PR↗
arXiv2010

The single ring theorem

We study the empirical measure $L_{A_n}$ of the eigenvalues of non-normal square matrices of the form $A_n=U_nD_nV_n$ with $U_n,V_n$ independent Haar distributed on the unitary group and $D_n$ real diagonal. We show that when the empirical measure of the eigenvalues of $D_n$ converges, and $D_n$ satisfies some technical conditions, $L_{A_n}$ converges towards a rotationally invariant measure on the complex plan whose support is a single ring. In particular, we provide a complete proof of Feinberg-Zee single ring theorem \cite{FZ}. We also consider the case where $U_n,V_n$ are independent Haar distributed on the orthogonal group.

math.PR↗
arXiv2009

Spectral measure of heavy tailed band and covariance random matrices

We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure $μ$ of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N by N symmetric matrix $Y_N^σ$ whose (i,j) entry is $σ(i/N,j/N)X_{ij}$ where $(X_{ij}, 0<i<j+1<\infty)$ is an infinite array of i.i.d real variables with common distribution in the domain of attraction of an $α$-stable law, $0<α<2$, and $σ$ is a deterministic function. For a random diagonal $D_N$ independent of $Y_N^σ$ and with appropriate rescaling $a_N$, we prove that the distribution $μ$ of $a_N^{-1}Y_N^σ+ D_N$ converges in mean towards a limiting probability measure which we characterize. As a special case, we derive and analyze the almost sure limiting spectral density for empirical covariance matrices with heavy tailed entries.

math.PR↗
arXiv2008

Asymptotics of unitary and othogonal matrix integrals

In this paper, we prove that in small parameter regions, arbitrary unitary matrix integrals converge in the large $N$ limit and match their formal expansion. Secondly we give a combinatorial model for our matrix integral asymptotics and investigate examples related to free probability and the HCIZ integral. Our convergence result also leads us to new results of smoothness of microstates. We finally generalize our approach to integrals over the othogonal group.

math.PR↗
arXiv2008

Regularization by free additive convolution, square and rectangular cases

The free convolution is the binary operation on the set of probability measures on the real line which allows to deduce, from the individual spectral distributions, the spectral distribution of a sum of independent unitarily invariant square random matrices or of a sum of free operators in a non commutative probability space. In the same way, the rectangular free convolution allows to deduce, from the individual singular distributions, the singular distribution of a sum of independent unitarily invariant rectangular random matrices. In this paper, we consider the regularization properties of these free convolutions on the whole real line. More specifically, we try to find continuous semigroups $(μ_t)$ of probability measures such that $μ_0$ is the Dirac mass at zero and such that for all positive $t$ and all probability measure $ν$, the free convolution of $μ_t$ with $ν$ (or, in the rectangular context, the rectangular free convolution of $μ_t$ with $ν$) is absolutely continuous with respect to the Lebesgue measure, with a positive analytic density on the whole real line. In the square case, we prove that in semigroups satisfying this property, no measure can have a finite second moment, and we give a sufficient condition on semigroups to satisfy this property, with examples. In the rectangular case, we prove that in most cases, for $μ$ in a continuous rectangular-convolution-semigroup, the rectangular convolution of $μ$ with $ν$ either has an atom at the origin or doesn't put any mass in a neighborhood of the origin, thus the expected property does not hold. However, we give sufficient conditions for analyticity of the density of the rectangular convolution of $μ$ with $ν$ except on a negligible set of points, as well as existence and continuity of a density everywhere.

math.PR↗
arXiv2007

Second order asymptotics for matrix models

We study several-matrix models and show that when the potential is convex and a small perturbation of the Gaussian potential, the first order correction to the free energy can be expressed as a generating function for the enumeration of maps of genus one. In order to do that, we prove a central limit theorem for traces of words of the weakly interacting random matrices defined by these matrix models and show that the variance is a generating function for the number of planar maps with two vertices with prescribed colored edges.

math.PR↗
arXiv2007

The spectrum of heavy-tailed random matrices

Let $X_N$ be an $N\ts N$ random symmetric matrix with independent equidistributed entries. If the law $P$ of the entries has a finite second moment, it was shown by Wigner \cite{wigner} that the empirical distribution of the eigenvalues of $X_N$, once renormalized by $\sqrt{N}$, converges almost surely and in expectation to the so-called semicircular distribution as $N$ goes to infinity. In this paper we study the same question when $P$ is in the domain of attraction of an $α$-stable law. We prove that if we renormalize the eigenvalues by a constant $a_N$ of order $N^{\frac{1}α}$, the corresponding spectral distribution converges in expectation towards a law $μ_α$ which only depends on $α$. We characterize $μ_α$ and study some of its properties; it is a heavy-tailed probability measure which is absolutely continuous with respect to Lebesgue measure except possibly on a compact set of capacity zero.

math.PR↗
arXiv2006

Limiting dynamics for spherical models of spin glasses at high temperature

We analyze the coupled non-linear integro-differential equations whose solutions is the thermodynamical limit of the empirical correlation and response functions in the Langevin dynamics for spherical p-spin disordered mean-field models. We provide a mathematically rigorous derivation of their FDT solution (for the high temperature regime) and of certain key properties of this solution, which are in agreement with earlier derivations based on physical grounds.

math.PR↗