Gérard Ben Arous

Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices

AbstractWe compute the limiting distributions of the largest eigenvalue of a complex Gaussian samplecovariance matrix when both the number of samples and the number of variables in each samplebecome large. When all but finitely many, say r, eigenvalues of the covariance matrix arethe same, the dependence of the limiting distribution of the largest eigenvalue of the samplecovariance matrix on those distinguished r eigenvalues of the covariance matrix is completelycharacterized in terms of an infinite sequence of new distribution functions that generalizethe Tracy-Widom distributions of the random matrix theory. Especially a phase transitionphenomena is observed. Our results also apply to a last passage percolation model and aqueuing model. 1 Introduction Consider M independent, identically distributed samples y 1 ,...,~y M , all of which are N ×1 columnvectors. We further assume that the sample vectors ~y k are Gaussian with mean µ and covarianceΣ, where Σ is a fixed N ×N positive matrix; the density of a sample ~y isp(~y) =1(2π)

Flots et series de Taylor stochastiques

SummaryWe study the expansion of the solution of a stochastic differential equation as an (infinite) sum of iterated stochastic (Stratonovitch) integrals. This enables us to give a universal and explicit formula for any invariant diffusion on a Lie group in terms of Lie brackets, as well as a universal and explicit formula for the brownian motion on a Riemannian manifold in terms of derivatives of the curvature tensor. The first of these formulae contains, and extends to the non nilpotent case, the results of Doss [6], Sussmann [17], Yamato [18], Fliess and Normand-Cyrot [7], Krener and Lobry [19] and Kunita [11] on the representation of solutions of stochastic differential equations.

Methods de laplace et de la phase stationnaire sur l'espace de wiener

We obtain, using large deviations principles, full asymptotic expansions of functionals of Laplace type on Wiener space for general (i.e. degenerate) diffusions extending the results of Schilder [12], Azencott [2] and Doss [7, 20]. The variational hypothesis (non degeneracy of the minima) used here is shown to be optimal. The first term of the expansion is explicitly computed. Using the integration by parts of Malliavin calculus the stationary phase method is also developed. The results of this paper are the basic fact used (in [5]) to obtain the asymptotic expansion for small time of the density of a degenerate diffusion, they are also relevant for semi-classical expansions

Current fluctuations for TASEP: A proof of the Prähofer–Spohn conjecture

We consider the family of two-sided Bernoulli initial conditions for TASEP which, as the left and right densities (p-, ρ+) are varied, give rise to shock waves and rarefaction fans—the two phenomena which are typical to TASEP. We provide a proof of Conjecture 7.1 of [Progr. Probab. 51 (2002) 185-204] which characterizes the order of and scaling functions for the fluctuations of the height function of two-sided TASEP in terms of the two densities ρ-, ρ+ and the speed y around which the height is observed. In proving this theorem for TASEP, we also prove a fluctuation theorem for a class of comer growth processes with external sources, or equivalently for the last passage time in a directed last passage percolation model with two-sided boundary conditions: ρ- and 1 - p + . We provide a complete characterization of the order of and the scaling functions for the fluctuations of this models last passage time L(N, M) as a function of three parameters: the two boundary/source rates p- and 1 - p + , and the scaling ratio y 2 = M/N. The proof of this theorem draws on the results of [Comm. Math. Phys. 265 (2006) 1-44] and extensively on the work of [Ann. Probab. 33 (2005) 1643-1697] on finite rank perturbations of Wishart ensembles in random matrix theory.

Complexity of random smooth functions on the high-dimensional sphere

We analyze the landscape of general smooth Gaussian functions on the sphere in dimension N, when N is large. We give an explicit formula for the asymptotic complexity of the mean number of critical points of finite and diverging index at any level of energy and for the mean Euler characteristic of level sets. We then find two possible scenarios for the bottom landscape, one that has a layered structure of critical values and a strong correlation between indexes and critical values and another where even at levels below the limiting ground state energy the mean number of local minima is exponentially large. We end the paper by discussing how these results can be interpreted in the language of spin glasses models.

Poisson convergence for the largest eigenvalues of heavy tailed random matrices

On etudie la loi des plus grandes valeurs propres de matrices aleatoires symetriques reelles et de covariance empirique quand les coefficients des matrices sont a queue lourde. On etend le resultat obtenu par Soshnikov dans (Electron. Commun. Probab. 9 (2004) 82-91) et on montre que le comportement asymptotique des plus grandes valeurs propres est determine par les plus grandes entrees de la matrice.

AGING IN THE RANDOM ENERGY MODEL

The random energy model (REM) has become a key reference model for glassy systems. In particular, it is expected to provide a prime example of a system whose dynamics shows aging, a universal phenomenon characterizing the dynamics of complex systems. The analysis of its activated dynamics is based on so-called trap models, introduced by Bouchaud, that are also used to mimic the dynamics of more complex disordered systems. In this Letter we report the first results that justify rigorously the trap model predictions in the REM.

Biased random walks on Galton–Watson trees with leaves

We consider a biased random walk

The Poisson kernel for certain degenerate elliptic operators

X_n

Methode de laplace: etude variationnelle des fluctuations de diffusions de type

on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant