Sharp mean-field estimates for the repulsive log gas in any dimension

TL;DR

This paper establishes precise mean-field estimates for a system of particles with repulsive logarithmic interactions in any dimension, improving the understanding of their collective behavior and partition function bounds.

Contribution

It provides sharp, uniform bounds on the partition function for the log gas in arbitrary dimensions, using a novel approach inspired by quantum field theory techniques.

Findings

Partition function is uniformly bounded in particle number N.

Logarithmic improvement in mean-field closeness estimates.

Method applicable to any dimension with repulsive log interactions.

Abstract

We prove sharp estimates for the mean-field limit of weakly interacting diffusions with repulsive logarithmic interaction in arbitrary dimension. More precisely, we show that the associated partition function is uniformly bounded in the number of particles $N$ for an arbitrary bounded base measure. Combined with the modulated free energy method, this amounts to a logarithmic improvement in $N$ of the current best available closeness estimates in the literature. Our arguments are inspired by and borrow ideas from Nelson's classical construction of the $\varphi^4_2$ Euclidean quantum field theory.