Confinement, Turbulence and Diffraction Catastrophes

Many features of large N_c transition that occurs in the spectral density of Wilson loops as a function of loop area (observed recently in numerical simulations of Yang-Mills theory by Narayanan and Neuberger) can be captured by a simple Burgers equation used to model turbulence. Spectral shock waves that precede this asymptotic limit exhibit universal scaling with N_c, with indices that can be related to Berry indices for diffraction catastrophes.Comment: Presented at PANIC 200

A multi-layer extension of the stochastic heat equation

Motivated by recent developments on solvable directed polymer models, we define a 'multi-layer' extension of the stochastic heat equation involving non-intersecting Brownian motions.Comment: v4: substantially extended and revised versio

On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures

This note, mostly expository, is devoted to Poincar{\'e} and log-Sobolev inequalities for a class of Boltzmann-Gibbs measures with singular interaction. Such measures allow to model one-dimensional particles with confinement and singular pair interaction. The functional inequalities come from convexity. We prove and characterize optimality in the case of quadratic confinement via a factorization of the measure. This optimality phenomenon holds for all beta Hermite ensembles including the Gaussian unitary ensemble, a famous exactly solvable model of random matrix theory. We further explore exact solvability by reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting the Hermite-Lassalle orthogonal polynomials as a complete set of eigenfunctions. We also discuss the consequence of the log-Sobolev inequality in terms of concentration of measure for Lipschitz functions such as maxima and linear statistics.Comment: Minor improvements. To appear in Geometric Aspects of Functional Analysis -- Israel Seminar (GAFA) 2017-2019", Lecture Notes in Mathematics 225

No multiple collisions for mutually repelling Brownian particles

Brownian particles in electrostatic interaction may pairwise collide when the interaction parameter is small. But multiple collisions are never possible

Brownian particles with electrostatic repulsion on the circle: Dyson's model for unitary random matrices revisited

The Brownian motion model introduced by Dyson [7] for the eigenvalues of unitary random matrices N x N is interpreted as a system of N interacting Brownian particles on the circle with electrostatic inter-particles repulsion. The aim of this paper is to define the finite particle system in a general setting including collisions between particles. Then, we study the behaviour of this system when the number of particles N goes to infinity (through the empirical measure process). We prove that a limiting measure-valued process exists and is the unique solution of a deterministic second-order PDE. The uniform law on [-π;π] is the only limiting distribution of µt when t goes to infinity and µt has an analytical density

A non linear stochastic differential equation involving Hilbert Transform

We consider a non-linear stochastic differential equation which involves the Hilbert transform, Xt=σ·Bt+2λ ∫t0 Imageu(s, Xs) ds. In the previous equation, u(t, ·) is the density of μt, the lax of Xt, and Image represents the Hilbert transform in the space variable. In order to define correctly the solutions, we first study the associated non-linear second-order integro-partial differential equation which can be reduced to the holomorphic Burgers equation. The real analyticity of solutions allows us to prove existence and uniqueness of the non-linear diffusion process. This stochastic differential equation has been introduced when studying the limit of systems of Brownian particles with electrostatic repulsion when the number of particles increases to infinity. More precisely, it has been show that the empirical measure process tends to the unique solution μ=(μt)tgreater-or-equal, slanted0 of the non-linear second-order integro-partial differential, equation studied here