Coulomb gas ensembles and Laplacian growth

TL;DR

This paper studies the behavior of eigenvalues in normal matrix ensembles under various potentials, showing they form a compact spectral droplet, and explores how adding dimensions affects this eigenvalue distribution.

Contribution

It introduces the concept of spectral droplets for eigenvalues in normal matrix ensembles and analyzes their evolution when the system's dimension increases.

Findings

Eigenvalues condense on a compact spectral droplet in the plane.

The spectral droplet's shape and properties depend on the confining potential.

Adding an extra dimension causes a predictable evolution of the eigenvalue distribution.

Abstract

We consider the normal matrix ensemble under a general confining potential. We find that the eigenvalues condensate on a compact set in the plane, which we call the spectral droplet. We also study the evolution of incrementally adding a dimension, i.e., adding an extra electron in this fermionic model.