Coulomb and Riesz gases: The known and the unknown

TL;DR

This paper reviews the mathematical properties of Coulomb and Riesz gases, focusing on defining their infinite point processes, understanding their equilibrium states, and discussing open problems and phase transitions.

Contribution

It provides a comprehensive overview of known results and open questions regarding the mathematical structure of Coulomb and Riesz gases, especially in the long-range interaction case.

Findings

Well-understood in the short-range case $s>d$

Open problems in defining infinite processes for $s<d$

Discussion of phase transition expectations

Abstract

We review what is known, unknown and expected about the mathematical properties of Coulomb and Riesz gases. Those describe infinite configurations of points in $\mathbb{R}^d$ interacting with the Riesz potential $\pm |x|^{-s}$ (resp. $-\log|x|$ for $s=0$). Our presentation follows the standard point of view of statistical mechanics, but we also mention how these systems arise in other important situations (e.g. in random matrix theory). The main question addressed in the article is how to properly define the associated infinite point process and characterize it using some (renormalized) equilibrium equation. This is largely open in the long range case $s<d$. For the convenience of the reader we give the detail of what is known in the short range case $s>d$. In the last part we discuss phase transitions and mention what is expected on physical grounds.