Concentration inequality around the thermal equilibrium measure of Coulomb gases

TL;DR

This paper establishes an optimal concentration inequality for Coulomb gases at intermediate temperatures, showing the empirical measure closely approximates the thermal equilibrium measure with high probability.

Contribution

It introduces a new concentration inequality for Coulomb gases at intermediate temperatures and develops functional inequalities to compare different measure norms without compact support.

Findings

Empirical measure concentrates around thermal equilibrium with probability exponentially close to 1.

Concentration inequality proven to be optimal in certain cases.

New functional inequalities relate Lipschitz and $H^{-1}$ norms for non-compact measures.

Abstract

This article deals with Coulomb gases at an intermediate temperature regime, in which no structure is observed at the microscopic level, but the mass in confined to a compact set. Our main result is a concentration inequality around the thermal equilibrium measure, stating that with probability exponentially close to $1,$ the empirical measure is $\mathcal{O}\left( \frac{1}{N^{\frac{1}{d}}}\right)$ close to the thermal equilibrium measure. We also prove that this concentration inequality is optimal in some sense. The main new tool are functional inequalities that allow us to compare the bounded Lipschitz norm of a measure to its $H^{-1}$ norm in some cases when the measure does not have compact support.