# Critical behavior at the localization transition on random regular graphs

## TL;DR

This paper investigates the critical behavior of the Anderson localization transition on random regular graphs, accurately determining the critical disorder and analyzing the correlation length divergence with notable corrections to scaling.

## Contribution

It provides the first precise estimate of the critical disorder and characterizes the critical divergence of the correlation length on random regular graphs.

## Key findings

- Critical disorder W_c = 18.17 ± 0.01 was determined.
- Correlation length diverges as (W_c - W)^(-1/2) near the transition.
- Pronounced corrections to scaling similar to high-dimensional and many-body localization models.

## Abstract

We study numerically the critical behavior at the localization transition in the Anderson model on infinite Bethe lattice and on random regular graphs. The focus is on the case of coordination number $m+1 = 3$, with a box distribution of disorder and in the middle of the band (energy $E=0$), which is the model most frequently considered in the literature. As a first step, we carry out an accurate determination of the critical disorder, with the result $W_c =18.17\pm 0.01$. After this, we determine the dependence of the correlation volume $N_\xi = m^\xi$ (where $\xi$ is the associated correlation length) on disorder $W$ on the delocalized side of the transition, $W < W_c$, by means of population dynamics. The asymptotic critical behavior is found to be $\xi \propto (W_c-W)^{-1/2}$, in agreement with analytical prediction. We find very pronounced corrections to scaling, in similarity with models in high spatial dimensionality and with many-body localization transitions.