Equilibrium Measures for a Class of Potentials with Discrete Rotational Symmetries

Abstract

The asymptotic analysis of the eigenvalue distribution of \(N\times N\) random normal matrix models in the large \(N\) limit naturally leads to a logarithmic energy problem with external potential in the complex plane. In the present paper, we consider this variational problem for the class of matrix models whose associated external potential is of the special form \(|z|^{2n}+tz^d+\bar{t}\bar{z}^d\), where \(n\) and \(d\) are positive integers satisfying \(d\le 2n\). By exploiting the discrete rotational invariance of such potentials, a simple symmetry reduction procedure is used to calculate the equilibrium measure for all admissible values of \(n,d\), and \(t\). It is shown that, for fixed \(n\) and \(d\), there is a critical value \(|t|=t_\mathrm{cr}\) such that the support of the equilibrium measure is simply connected for \(|t| t_\mathrm{cr}\).