Equilibrium measures for a class of potentials with discrete rotational symmetries

Abstract

In this note the logarithmic energy problem with external potential $|z{|}^{2n}+t{z}^d+\bar{t}{\bar{z}}^d$ is considered in the complex plane, where $n$ and $d$ are positive integers satisfying $d\le 2n$. Exploiting the discrete rotational invariance of the potential, 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 $|t|=t_{cr}$ such that the support of the equilibrium measure is simply connected for $|t|<t_{cr}$ and has $d$ connected components for $|t|>t_{cr}$.

23 pages, 3 figures

References in corpus (3)

• Normal random matrix ensemble as a growth problem
• Density of eigenvalues of random normal matrices
• Normal matrix models, dbar-problem, and orthogonal polynomials on the complex plane