# Level compressibility for the Anderson model on regular random graphs and the eigenvalue statistics in the extended phase

## TL;DR

This paper investigates the spectral properties of the Anderson model on large regular random graphs, showing that energy levels exhibit Wigner-Dyson statistics in the extended phase near the transition point, based on numerical solutions.

## Contribution

It provides the first numerical evidence that level compressibility approaches zero in the extended phase of the Anderson model on regular random graphs, indicating Wigner-Dyson statistics.

## Key findings

- Level compressibility approaches zero in the extended phase.
- Energy levels follow Wigner-Dyson statistics near the transition.
- Results are consistent with predictions for Erdős-Rényi graphs.

## Abstract

We calculate the level compressibility $\chi(W,L)$ of the energy levels inside $[-L/2,L/2]$ for the Anderson model on infinitely large random regular graphs with on-site potentials distributed uniformly in $[-W/2,W/2]$. We show that $\chi(W,L)$ approaches the limit $\lim_{L \rightarrow 0^+} \chi(W,L) = 0$ for a broad interval of the disorder strength $W$ within the extended phase, including the region of $W$ close to the critical point for the Anderson transition. These results strongly suggest that the energy levels follow the Wigner-Dyson statistics in the extended phase, consistent with earlier analytical predictions for the Anderson model on an Erd\"os-R\'enyi random graph. Our results are obtained from the accurate numerical solution of an exact set of equations valid for infinitely large regular random graphs.