Quenched large deviations for Monte Carlo integration with Coulomb gases

arXiv:2508.01392v2 Announce Type: replace 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 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.

Source: arXiv cs.LG | 2026-06-29