Computation of Power Law Equilibrium Measures on Balls of Arbitrary Dimension

TL;DR

This paper introduces an efficient numerical method for computing power law equilibrium measures in any dimension, leveraging new recurrence relations for radial Jacobi polynomials, and demonstrates its accuracy and potential for exploring complex support structures.

Contribution

The paper develops a novel recursive approach for radial Jacobi polynomials on d-dimensional balls, enabling dimension-independent computation of equilibrium measures.

Findings

Method achieves high accuracy in numerical experiments.

Computational complexity is independent of dimension d.

Method outperforms particle swarm simulations in high dimensions.

Abstract

We present a numerical approach for computing attractive-repulsive power law equilibrium measures in arbitrary dimension. We prove new recurrence relationships for radial Jacobi polynomials on $d$-dimensional ball domains, providing a substantial generalization of the work based on recurrence relationships of Riesz potentials on arbitrary dimensional balls. Among the attractive features of the numerical method are good efficiency due to recursively generated banded and approximately banded Riesz potential operators and computational complexity independent of the dimension $d$, in stark contrast to the widely used particle swarm simulation approaches for these problems which scale catastrophically with the dimension. We present several numerical experiments to showcase the accuracy and applicability of the method and discuss how our method compares with alternative numerical approaches and conjectured analytical solutions which exist for certain special cases. Finally, we discuss how our method can be used to explore the analytically poorly understood gap formation boundary to spherical shell support.