Edge scaling limit of the spectral radius for random normal matrix ensembles at hard edge

TL;DR

This paper studies the behavior of the spectral radius in a random normal matrix model with eigenvalues constrained to a droplet, revealing exponential fluctuations at the hard edge for large matrices.

Contribution

It establishes the edge scaling limit of the spectral radius for radially symmetric external fields, showing exponential fluctuation behavior as the matrix size grows.

Findings

Spectral radius fluctuations follow an exponential distribution at the hard edge.

Order statistics of eigenvalue moduli are derived.

Results apply to large matrix limits with radially symmetric external fields.

Abstract

We investigate a random normal matrix model with eigenvalues forced to be in the droplet, the support of the equilibrium measure associated with an external field. For radially symmetric external fields, we show that the fluctuations of the spectral radius around a hard edge tend to follow an exponential distribution as the number of eigenvalues tends to infinity. As a corollary, we obtain the order statistics of the moduli of eigenvalues.