Matrix models and eigenvalue statistics for truncations of classical ensembles of random unitary matrices

TL;DR

This paper develops matrix models for truncated classical unitary ensembles, enabling the analysis of eigenvalue distributions across all beta values, with applications to quantum resonances and electrostatics.

Contribution

It introduces sparse matrix models that replicate the spectral measures of truncated classical ensembles for all beta, extending previous results limited to specific cases.

Findings

Derived joint eigenvalue laws for truncated ensembles

Applicable to all beta > 0, not just 1, 2, 4

Provides insights into quantum resonances and electrostatics

Abstract

We consider random non-normal matrices constructed by removing one row and column from samples from Dyson's circular ensembles or samples from the classical compact groups. We develop sparse matrix models whose spectral measures match these ensembles. This allows us to compute the joint law of the eigenvalues, which have a natural interpretation as resonances for open quantum systems or as electrostatic charges located in a dielectric medium. Our methods allow us to consider all values of $\beta>0$, not merely $\beta=1,2,4$.