Non-Archimedean Coulomb gases

Abstract

This article aims to study the Coulomb gas model over the d-dimensional p-adic space. We establish the existence of equilibrium measures and the Γ-limit for the Coulomb energy functional when the number of configurations tends to infinity. For a cloud of charged particles confined into the unit ball, we compute the equilibrium measure and the minimum of its Coulomb energy functional. The p-adic Coulomb energy is the continuum limit of the minus a hierarchical Hamiltonian attached to a spin glass model with a p-adic coupling.