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.

Minor revision, accepted for publication in Analysis and Mathematical Physics

References in corpus (4)

• A probabilistic approach to the geometry of the \ell_p^n-ball
• Computing Equilibrium Measures with Power Law Kernels
• On the solution of a Riesz equilibrium problem and integral identities for special functions
• Computation of Power Law Equilibrium Measures on Balls of Arbitrary Dimension

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• Explicit minimisers for anisotropic Riesz energies