Anderson localization and ergodicity on random regular graphs

A numerical study of Anderson transition on random regular graphs (RRG) 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 certain sense, the RRG ensemble can be seen as infinite-dimensional ($d\to\infty$) cousin of 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 small ($N\ll N_c$) to large ($N\gg N_c$) system, where $N_c$ is the correlation volume diverging exponentially at the transition. A distinct feature of this crossover is a nonmonotonicity of the spectral and wavefunction statistics, which is related to properties of the critical phase in the studied model and renders the finite-size analysis highly non-trivial. Our results support an analytical prediction that states in the delocalized phase (and at $N\gg N_c$) are ergodic in the sense that their inverse participation ratio scales as $1/N$.