Generic critical points of normal matrix ensembles

TL;DR

This paper explores the critical points of normal matrix ensembles, linking their geometric evolution to Laplacian Growth singularities and describing the scaling behavior with Painlevé equations.

Contribution

It establishes a connection between critical points in matrix ensembles and complex curve singularities, providing a new analytical framework for their study.

Findings

The evolution of complex curves at critical points relates to Laplacian Growth singularities.

Scaling behavior at critical points is described by the first Painlevé transcendent.

Discretization regularizes the complex curve, affecting its geometric properties.

Abstract

The evolution of the degenerate complex curve associated with the ensemble at a generic critical point is related to the finite time singularities of Laplacian Growth. It is shown that the scaling behavior at a critical point of singular geometry $x^3 \sim y^2$ is described by the first Painlev\'e transcendent. The regularization of the curve resulting from discretization is discussed.