Generalized transport inequalities and concentration bounds for Riesz-type gases

TL;DR

This paper establishes generalized transport inequalities for Riesz-type gases, linking energy and measure norms, and derives concentration bounds and Moser-Trudinger inequalities for these systems.

Contribution

It introduces a unified framework connecting Riesz energy with measure norms, extending Coulomb inequalities, and deriving new concentration and functional inequalities.

Findings

Derived generalized Coulomb transport inequalities.

Proved measure concentration around equilibrium states.

Established Moser-Trudinger-type inequalities for fluctuations.

Abstract

This paper explores the connection between a generalized Riesz electric energy and norms on the set of probability measures defined in terms of duality. We derive functional inequalities linking these two notions, recovering and generalizing existing Coulomb transport inequalities. We then use them to prove concentration of measure around the equilibrium and thermal equilibrium measures. Finally, we leverage these concentration inequalities to obtain Moser-Trudinger-type inequalities, which may also be interpreted as bounds on the Laplace transform of fluctuations.