# Non-Archimedean Coulomb Gases

## TL;DR

This paper investigates Coulomb gases in p-adic spaces, establishing equilibrium measures and energy limits, and connecting Coulomb energy with hierarchical spin glass models in a novel non-Archimedean setting.

## Contribution

It introduces the study of Coulomb gases over p-adic spaces, proving existence of equilibrium measures and deriving the Gamma-limit of Coulomb energy functional.

## Key findings

- Existence of equilibrium measures in p-adic Coulomb gases.
- Explicit computation of equilibrium measure for particles in the unit ball.
- Connection between Coulomb energy and hierarchical spin glass Hamiltonian.

## Abstract

This article aims to study the Coulomb gas model over the $d$-dimensional $p$-adic space. We establish the existence of equilibria measures and the $\Gamma$-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. In the $p$-adic setting the Coulomb energy is the continuum limit of the minus a hierarchical Hamiltonian attached to a spin glass model with a $p$-adic coupling.