Threshold condensation to singular support for a Riesz equilibrium problem

TL;DR

This paper analyzes the equilibrium measure for Riesz s-kernel interactions with external fields, revealing a threshold phenomenon where the measure condenses onto a singular set, especially in the logarithmic case.

Contribution

It introduces a detailed analysis of the equilibrium measure's structure, showing a phase transition between continuous and singular parts depending on the external field's power.

Findings

Equilibrium measure can be a mixture of continuous and singular parts.

A threshold phenomenon causes dimension reduction or condensation on the singular part.

In the logarithmic case, condensation occurs on a sphere of specific radius.

Abstract

We compute the equilibrium measure in dimension d=s+4 associated to a Riesz s-kernel interaction with an external field given by a power of the Euclidean norm. Our study reveals that the equilibrium measure can be a mixture of a continuous part and a singular part. Depending on the value of the power, a threshold phenomenon occurs and consists of a dimension reduction or condensation on the singular part. In particular, in the logarithmic case s=0 (d=4), there is condensation on a sphere of special radius when the power of the external field becomes quadratic. This contrasts with the case d=s+3 studied previously, which showed that the equilibrium measure is fully dimensional and supported on a ball. Our approach makes use, among other tools, of the Frostman or Euler-Lagrange variational characterization, the Funk-Hecke formula, the Gegenbauer orthogonal polynomials, and hypergeometric special functions.