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Regular version of the site

Article

Anderson localization and ergodicity on random regular graphs

Physical Review B: Condensed Matter and Materials Physics. 2016. Vol. 94. P. 220203-1-220203-6.
Tikhonov K., Mirlin A. D., Skvortsov M. A.

A numerical study of Anderson transition on random regular graphs (RRGs) with diagonal disorder is performed.
The problem can be described as a tight-binding model on a lattice with N sites that is locally a tree with constant connectivity.
In a certain sense, the RRG ensemble can be seen as an infinite-dimensional (d→∞) cousin of the Anderson model in
d dimensions. We focus on the delocalized side of the transition and stress the importance of finite-size effects.
We show that the data can be interpreted in terms of the finite-size crossover from a small (N >> Nc) to a large (N >> Nc) system, where Nc is the correlation volume diverging exponentially at the transition. A distinct feature of this crossover is a nonmonotonicity of the spectral and wave-function statistics, which is related to properties of the critical phase in the studied model and renders the finite-size analysis highly nontrivial. Our results support an analytical prediction that states in the delocalized phase (and at N >> Nc) are ergodic in the sense that their inverse participation ratio scales as 1/N