# Dynamics of a planar Coulomb gas

## TL;DR

This paper analyzes the long-term behavior of a planar Coulomb gas with non-convex interactions, establishing well-posedness, Poincaré inequalities, and identifying regimes through second moment dynamics, including connections to Cox-Ingersoll-Ross processes.

## Contribution

It demonstrates well-posedness and functional inequalities for a non-convex planar Coulomb gas, and links second moment dynamics to Cox-Ingersoll-Ross processes, revealing new regimes.

## Key findings

- System is well-posed for any inverse temperature.
- Poincaré inequalities hold despite non-convex interactions.
- Second moment dynamics follow a Cox-Ingersoll-Ross process.

## Abstract

We study the long-time behavior of the dynamics of interacting planar Brow-nian particles, confined by an external field and subject to a singular pair repulsion. The invariant law is an exchangeable Boltzmann -- Gibbs measure. For a special inverse temperature, it matches the Coulomb gas known as the complex Ginibre ensemble. The difficulty comes from the interaction which is not convex, in contrast with the case of one-dimensional log-gases associated with the Dyson Brownian Motion. Despite the fact that the invariant law is neither product nor log-concave, we show that the system is well-posed for any inverse temperature and that Poincar{\'e} inequalities are available. Moreover the second moment dynamics turns out to be a nice Cox -- Ingersoll -- Ross process in which the dependency over the number of particles leads to identify two natural regimes related to the behavior of the noise and the speed of the dynamics.