Threshold condensation to singular support for a Riesz equilibrium problem

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.