Vortex patch equilibria of the Euler equation and random normal matrices

Abstract

A new analogy is drawn between vortex patch, or V-state, equilibria of the Euler equations for ideal fluids and the planar limit of random normal matrix models. Physically the former is a quite different fluid dynamical problem to Hele-Shaw flow, or Laplacian growth, to which an analogy with matrix models has become well known in recent years. The connection of random matrices with vortex dynamics is made via the so-called modified Schwarz potential. This theoretical link, while interesting in itself, has immediate ramifications for random matrix theory by virtue of a transfer of mathematical technology already well developed for vortex dynamics. Hence for multi-support matrix models in the planar limit we describe a constructive approach to the inverse problem of finding the shapes of the eigenvalue supports for a given potential using automorphic conformal mappings within a Schottky model of the underlying algebraic curves and use of the Schottky–Klein prime function. Two-support eigenvalue distributions in a quartic potential are given as an illustrative example.