Quenched large deviations for Monte Carlo integration with Coulomb gases

TL;DR

This paper demonstrates that using large deviations principles, a Monte Carlo approximation of the potential in Coulomb gas-based Gibbs measures preserves the efficiency of the integration algorithm, even with minimal assumptions.

Contribution

It extends large deviations analysis to Monte Carlo approximations of potentials in Coulomb gases, ensuring reliable integration performance.

Findings

Random potential approximation maintains large deviation principles.

Minimal assumptions suffice for non-singular kernels.

Uniform convergence of potential approximation is established for Coulomb interactions.

Abstract

Gibbs measures, such as Coulomb gases, are popular in modelling systems of interacting particles. Recently, we proposed to use Gibbs measures as randomized numerical integration algorithms with respect to a target measure $\pi$ on $\mathbb R^d$, following the heuristics that repulsiveness between particles should help reduce integration errors. A major issue in this approach is to tune the interaction kernel and confining potential of the Gibbs measure, so that the equilibrium measure of the system is the target distribution $\pi$. Doing so usually requires another Monte Carlo approximation of the \emph{potential}, i.e. the integral of the interaction kernel with respect to $\pi$. Using the methodology of large deviations from Garcia--Zelada (2019), we show that a random approximation of the potential preserves the fast large deviation principle that guarantees the proposed integration algorithm to outperform independent or Markov quadratures. For non-singular interaction kernels, we make minimal assumptions on this random approximation, which can be the result of a computationally cheap Monte Carlo preprocessing. For the Coulomb interaction kernel, we need the approximation to be based on another Gibbs measure, and we prove in passing a control on the uniform convergence of the approximation of the potential.