Alice Guionnet

On the stability of interacting processes with applications to filtering and genetic algorithms

Abstract The stability properties of a class of interacting measure valued processes arising in nonlinear filtering and genetic algorithm theory is discussed. Simple sufficient conditions are given for exponential decays. These criteria are applied to study the asymptotic stability of the nonlinear filtering equation and infinite population models as those arising in Biology and evolutionary computing literature. On the basis of these stability properties we also propose a uniform convergence theorem for the interacting particle numerical scheme of the nonlinear filtering equation introduced in a previous work. In the last part of this study we propose a refinement genetic type particle method with periodic selection dates and we improve the previous uniform convergence results. We finally discuss the uniform convergence of particle approximations including branching and random population size systems.

Aging of spherical spin glasses

Abstract. Sompolinski and Zippelius (1981) propose the study of dynamical systems whose invariant measures are the Gibbs measures for (hard to analyze) statistical physics models of interest. In the course of doing so, physicists often report of an “aging” phenomenon. For example, aging is expected to happen for the Sherrington-Kirkpatrick model, a disordered mean-field model with a very complex phase transition in equilibrium at low temperature. We shall study the Langevin dynamics for a simplified spherical version of this model. The induced rotational symmetry of the spherical model reduces the dynamics in question to an N-dimensional coupled system of Ornstein-Uhlenbeck processes whose random drift parameters are the eigenvalues of certain random matrices. We obtain the limiting dynamics for N approaching infinity and by analyzing its long time behavior, explain what is aging (mathematically speaking), what causes this phenomenon, and what is its relationship with the phase transition of the corresponding equilibrium invariant measures.

Large Random Matrices: Lectures on Macroscopic Asymptotics

Wigner matrices and moments estimates.- Wigner#x2019 s theorem.- Wigners matrices more moments estimates.- Words in several independent Wigner matrices.- Wigner matrices and concentration inequalities.- Concentration inequalities and logarithmic Sobolev inequalities.- Generalizations.- Concentration inequalities for random matrices.- Matrix models.- Maps and Gaussian calculus.- First-order expansion.- Second-order expansion for the free energy.- Eigenvalues of Gaussian Wigner matrices and large deviations.- Large deviations for the law of the spectral measure of Gaussian Wigners matrices.- Large Deviations of the Maximum Eigenvalue.- Stochastic calculus.- Stochastic analysis for random matrices.- Large deviation principle for the law of the spectral measure of shifted Wigner matrices.- Asymptotics of Harish-Chandra-Itzykson-Zuber integrals and of Schur polynomials.- Asymptotics of some matrix integrals.- Free probability.- Free probability setting.- Freeness.- Free entropy.- Basics of matrices.- Basics of probability theory.

Large deviations for Langevin spin glass dynamics

We study the asymptotic behaviour of asymmetrical spin glass dynamics in a Sherrington-Kirkpatrick model as proposed by Sompolinsky-Zippelius. We prove that the annealed law of the empirical measure on path space of these dynamics satisfy a large deviation principle in the high temperature regime. We study the rate function of this large deviation principle and prove that it achieves its minimum value at a unique probability measureQ which is not markovian. We deduce that the quenched law of the empirical measure converges to δ Q . Extending then the preceeding results to replicated dynamics, we investigate the quenched behavior of a single spin. We get quenched convergence toQ in the case of a symmetric initial law and even potential for the free spin.

Large deviations upper bounds and central limit theorems for non-commutative functionals of gaussian large random matrices

Abstract We obtain large deviation upper bounds and central limit theorems for non-commutative functionals of large Gaussian band matrices and deterministic diagonal matrices with converging spectral measure. As a consequence, we derive such type of results for the spectral measure of Gaussian band matrices and Gaussian sample covariance matrices.

Large deviations and stochastic calculus for large random matrices

Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, operator theory, number theory, quantum field theory, string theory etc... In the last ten years, they attracted lots of interests, in particular due to a serie of mathematical breakthroughs allowing for instance a better understanding of local properties of their spectrum, answering universality questions, connecting these issues with growth processes etc. In this survey, we shall discuss the problem of the large deviations of the empirical measure of Gaussian random matrices, and more generally of the trace of words of independent Gaussian random matrices. We shall describe how such issues are motivated either in physics/combinatorics by the study of the so-called matrix models or in free probability by the definition of a non-commutative entropy. We shall show how classical large deviations techniques can be used in this context. These lecture notes are supposed to be accessible to non probabilists and non free-probabilists.

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.

On the stability of measure valued processes with applications to filtering

Abstract The purpose of this work is to present a novel approach to study the asymptotic stability of a class of measure valued processes arising in biology, in the theory of genetic type algorithms and in advanced signal processing, A new aspect of this result, as compared to existing literature on filtering, is that our approach is not based on auxiliary Hilbert projective metrics but only on Dobrushins ergodic coefficient. Another advantage is that it allows to treat time in-homogencous signals taking values in a polish space and it is not restricted to non-linear filtering settings. What also makes our results interesting and new is that these qualitative properties lead to our knownledge to a first uniform convergence result with respect to time for u class of interacting particle schemes.

Asymptotics of unitary and orthogonal matrix integrals

Abstract 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 orthogonal group.

Invariant beta ensembles and the Gauss-Wigner crossover.

We define a new diffusive matrix model converging toward the β-Dyson Brownian motion for all β is an element of [0,2] that provides an explicit construction of beta ensembles of random matrices that is invariant under the orthogonal or unitary group. For small values of β, our process allows one to interpolate smoothly between the Gaussian distribution and the Wigner semicircle. 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 semicircle.