Coulomb Gases Under Constraint: Some Theoretical and Numerical Results

Abstract

We consider Coulomb gas models for which the empirical measure typically concentrates, when the number of particles becomes large, on an equilibrium measure minimizing an electrostatic energy. We study the behavior when the gas is conditioned on a rare event. We first show that the special case of quadratic confinement and linear constraint is exactly solvable due to a remarkable factorization, and that the conditioning has then the simple effect of shifting the cloud of particles without deformation. To address more general cases, we perform a theoretical asymptotic analysis relying on a large deviations technique known as the Gibbs conditioning principle. The technical part amounts to establishing that the conditioning ensemble is an Icontinuity set of the energy. This leads to characterizing the conditioned equilibrium measure as the solution of a modified variational problem. For simplicity, we focus on linear statistics and on quadratic statistics constraints. Finally, we numerically illustrate our predictions and explore cases in which no explicit solution is known. For this, we use a Generalized Hybrid Monte Carlo algorithm for sampling from the conditioned distribution for a finite but large system. Contents 12 4.1. Description of the algorithm 12 4.2. Numerical results 17 Acknowledgements 20 Appendix A.

The forbidden region for random zeros: Appearance of quadrature domains

“…For further results pertaining to probabilities of rare events for stationary random processes, we refer to, see [4,5,8,20,50,63].…”

Quasi-stationary distribution for Hamiltonian dynamics with singular potentials

“…Remark 1.1. The long time behavior of the law of the process (1.5) has been investigated in [30,38,3,6] when γ and Σ are constant, and V is C ∞ on S (see also [18,19,27,41,47,10] and references therein).…”