Equilibrium measures for higher dimensional rotationally symmetric Riesz gases

TL;DR

This paper characterizes equilibrium measures for higher-dimensional Riesz gases with radially symmetric external fields, providing explicit densities, constructing associated potentials, and exploring applications to classical and Coulomb gases.

Contribution

It offers a novel inverse construction linking prescribed equilibrium densities to external potentials and derives explicit formulas using hypergeometric functions.

Findings

Explicit equilibrium densities for Riesz gases with power series radial profiles.

Derived external potentials for densities proportional to (1-|x|^2)^α, expressed via hypergeometric functions.

Identified conditions for equilibrium measures supported on boundary hyperplanes in Coulomb gases.

Abstract

We study equilibrium measures for Riesz gases in dimension $d$ with pairwise interaction kernel $|x-y|^{-s}$, subject to radially symmetric external fields. We characterise broad classes of confining potentials for which the equilibrium measure is supported on the unit ball and admits an explicit density. Our main contribution is a converse construction: starting from a prescribed radially symmetric equilibrium density given as a power series in the squared radius, we determine the associated external potential and establish the corresponding Euler-Lagrange variational conditions. A key ingredient in the proof is an identity between two ${}_3F_2$ hypergeometric functions evaluated at unit argument, which is of independent interest. As applications, we identify the external potentials corresponding to equilibrium densities proportional to $(1-|x|^2)^\alpha$, $\alpha>-1$, and show that these potentials can be expressed in terms of Gauss hypergeometric functions ${}_2F_1$, reducing to polynomials for special values of $\alpha$. We also determine the equilibrium measure associated with purely power-type external potentials, often referred to as Freud or Mittag--Leffler potentials in the context of log gases, for which the equilibrium density admits an explicit ${}_2F_1$ representation. Furthermore, we apply our framework to a Coulomb gas in dimension $d+1$ confined by a harmonic potential to the half-space. We derive a necessary condition under which the equilibrium measure is fully supported on the boundary hyperplane of dimension $d$, with the induced density corresponding to that of a Riesz gas with exponent $s=d-1$.