# How much can the eigenvalues of a random Hermitian matrix fluctuate?

## TL;DR

This paper investigates the maximum deviation of eigenvalues from their expected locations in large Hermitian random matrices, providing optimal rigidity estimates using advanced probabilistic and asymptotic techniques.

## Contribution

It introduces a novel approach combining extreme value theory, multiplicative chaos, and Hankel determinant analysis to derive optimal eigenvalue rigidity estimates.

## Key findings

- Established optimal eigenvalue rigidity bounds.
- Analyzed fractal structure of eigenvalue counting functions.
- Connected eigenvalue fluctuations with log-correlated processes.

## Abstract

The goal of this article is to study how much the eigenvalues of large Hermitian random matrices deviate from certain deterministic locations -- or in other words, to investigate optimal rigidity estimates for the eigenvalues. We do this in the setting of one-cut regular unitary invariant ensembles of random Hermitian matrices -- the Gaussian Unitary Ensemble being the prime example of such an ensemble. Our approach to this question combines extreme value theory of log-correlated stochastic processes, and in particular the theory of multiplicative chaos, with asymptotic analysis of large Hankel determinants with Fisher-Hartwig symbols of various types, such as merging jump singularities, size-dependent impurities, and jump singularities approaching the edge of the spectrum. In addition to optimal rigidity estimates, our approach sheds light on the fractal geometry of the eigenvalue counting function.