# Evidence of the Poisson/Gaudin-Mehta phase transition for banded matrices on global scales

## TL;DR

This paper demonstrates the existence of a phase transition in the spectral properties of symmetric banded matrices as their bandwidth varies, supported by theoretical proof and numerical experiments.

## Contribution

It proves the Poisson/Gaudin-Mehta phase transition at a critical bandwidth and introduces a new conjectured transition related to eigenvalue localization.

## Key findings

- Phase transition at bandwidth ~√N in level density moments
- Numerical evidence for a second transition at bandwidth ~ (2/5)N
- Identification of critical points in spectral behavior

## Abstract

We prove that the Poisson/Gaudin--Mehta phase transition conjectured to occur when the bandwidth of an $N \times N$ symmetric banded matrix grows like $\sqrt N$ is observable as a critical point in the fourth moment of the level density for a wide class of symmetric banded matrices. A second critical point when the bandwidth grows like ${2 \over 5} N$ leads to a new conjectured phase transition in the eigenvalue localization, whose existence we demonstrate in numerical experiments.