A large deviation principle for weighted Riesz interactions

TL;DR

This paper establishes a large deviation principle for empirical measures influenced by Riesz potential interactions on compact sets, under certain measure conditions, advancing understanding of probabilistic behavior in potential theory.

Contribution

It introduces a large deviation principle for Riesz interactions with new conditions on the underlying measures and sets, extending previous theoretical frameworks.

Findings

Valid for measures satisfying Bernstein-Markov property

Applicable to smooth submanifolds with mass-density conditions

Provides sufficient conditions for measure properties on compact sets

Abstract

We prove a large deviation principle for the sequence of push-forwards of empirical measures in the setting of Riesz potential interactions on compact subsets K in R^d with continuous external fields. Our results are valid for base measures on K satisfying a strong Bernstein-Markov type property for Riesz potentials. Furthermore, we give sufficient conditions on K (which are satisfied if K is a smooth submanifold) so that a measure on K which satisfies a mass-density condition will also satisfy this strong Bernstein-Markov property.