On the distribution of eigenvalues of GUE and its minors at fixed index

TL;DR

This paper provides bounds on the eigenvalue gaps of GUE matrices that are tight in the large size limit and applies these results to establish universality for GUE minors, impacting the study of random hives.

Contribution

It introduces bounds on eigenvalue gaps of GUE matrices that avoid logarithmic loss and proves fixed index universality for GUE minors, linking to random hive models.

Findings

Bounds on eigenvalue gaps without logarithmic loss

Fixed index universality for GUE minor process

Applications to random hive models

Abstract

We obtain bounds on the distribution of normalized gaps of eigenvalues of $N \times N$ GUE matrix in the bulk, that do not lose logarithmic factors of $N$ in the limit $N \to \infty$. As an application, we obtain fixed index universality results for the GUE minor process, which in turn are useful for establishing limiting results for random hives with GUE boundary data.