Uniform in time mean field limits for 1D Riesz gases

This note is based on a talk given at the Séminaire Laurent Schwartz in February 2024. The goal is to review a recent joint work with Arnaud Guillin (Université Clermont-Auvergne) and Pierre Monmarché (Sorbonne Université) published in Journal de l’École polytechnique - Mathématiques (2023) [GLM23]. Consider a one dimensional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> -particle system in singular repulsive mean field interaction. The main motivating example is the (generalized) Dyson Brownian motion which holds importance in Random Matrix Theory. We wish to show the convergence, as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> goes to infinity, of the empirical distribution of the system towards the solution of a non linear PDE. We describe a method that relies only on the well posedness of the system of particles and which provides a quantitative (and in some cases uniform in time) result. We make full use of the fact that in dimension one the particles will stay ordered, and that as a consequence the interaction we consider will be convex. Using a coupling method, we prove that by taking any independent sequence of empirical measures, it is a Cauchy sequence. Then, independence ensures the fact that the limit is an almost surely constant random variable which we then identify. This method requires in particular no study of the non linear limit.