. Motivated by the stochastic quantization approach to large N matrix models, we study solutions to free stochastic differential equations dX t = dS t \Gamma 1 2 f(X t )dt where S t is a free brownian motion. We show existence, uniqueness and Markov property of solutions. We define a relative free entropy as well as a relative free Fisher information, and show that these quantities behave as in the classical case. Finally we show that, in contrast with classical diffusions, in general the asymptotic distribution of the free diffusion does not converge, as t ! 1, towards the master field (i.e. the Gibbs state). 1. Introduction The purpose of this paper is to start the study of diffusion equations where the driving noise is a free brownian motion. Reasons for considering such equations will be explained in the next sections of this introduction. 1.1 Gibbs states and diffusion theory. Let V be a C 2 function on R d , with Z = Z R d e \GammaV (x) dx ! 1: The probability measur...

Abstract. We study the limiting spectral measure of large symmetric random matrices of linear algebraic structure. For Hankel and Toeplitz matrices generated by i.i.d. random variables {Xk} of unit variance, and for symmetric Markov matrices generated by i.i.d. random variables {Xi,j}j>i of zero mean and unit variance, scaling the eigenvalues by √ n we prove the almost sure, weak convergence of the spectral measures to universal, non-random, symmetric distributions γH, γM, and γT of unbounded support. The moments of γH and γT are the sum of volumes of solids related to Eulerian numbers, whereas γM has a bounded smooth density given by the free convolution of the semi-circle and normal densities. For symmetric Markov matrices generated by i.i.d. random variables {Xi,j}j>i of mean m and finite variance, scaling the eigenvalues by n we prove the almost sure, weak convergence of the spectral measures to the atomic measure at −m. If m = 0, and the fourth moment is finite, we prove that the spectral norm of Mn scaled by √ 2n log n converges almost surely to one. 1. Introduction and

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.

We use free probability techniques for computing spectra and Brown measures of some non hermitian operators in finite von Neumann algebras. Examples includeun+u1where unandu1are the generators of Zn and Z respectively, in the free product ZnZ, or elliptic elements, of the form S+iS where S and S are free semi-circular elements of variance and.

Abstract. We define a free probability analogue of the Wasserstein metric, which extends the classical one. In dimension one, we prove that the square of the Wasserstein distance to the semi-circle distribution is majorized by a modified free entropy quantity. 0

Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of An and Bn is obtained and studied.

Abstract. Based on a new analytical approach to the definition of additive free convolution on probability measures on the real line we prove free analogs of limit theorems for sums for non-identically distributed random variables in classical Probability Theory. 1.

We derive a moderate deviations principle for matrices of the form XN = DN + WN where WN are Wigner matrices and DN is a sequence of deterministic matrices whose spectral measures converge in a strong sense to a limit D . Our techniques are based on a dynamical approach introduced by Cabanal-Duvillard and Guionnet.

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We derive a moderate deviations principle for matrices of the form XN = DN + WN where WN are Wigner matrices and DN is a sequence of deterministic matrices whose spectral measures converge in a strong sense to a limit D . Our techniques are based on a dynamical approach introduced by Cabanal-Duvillard and Guionnet. 1