I will discuss Anderson (de-)localization on random regular graphs (RRG), which have locally the structure of a tree but do not have boundary (and thus possess large-scale loops). Our analytical treatment of the RRG model uses a field-theoretical supersymmetry approach and the saddle-point analysis justified by large “volume” (number of sites) $N$. The resulting saddle-point equation can be efficiently solved numerically by population dynamics, and also analyzed analytically [1]. The obtained results are in perfect agreement with those of exact diagonalization. In the delocalized phase on RRG, eigenfunctions are ergodic in the sense that their inverse participation ratio scales as $1/N$ and the spectral statistics is Wigner-Dyson in the large-$N$ limit. This limit is reached via a 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, $\ln N_c ~ \xi_c$, where $\xi_c$ is the correlation length. A distinct feature of this crossover is a non-monotonicity of the spectral and wave-function statistics, which is related to properties of the critical phase in the RRG model [2]. We have also peformed a detailed study of eigenfunction and energy level correlations, fully confirming the ergodicity of the delocalized phase in the large-$N$ limit [1]. We further demonstrate numerically that the correlation length on the delocalized side, $\xi_c$, diverges with the critical index $ν_d = 1/2$, in agreement with analytical result [3]. Importantly, properties of the RRG model differ crucially from those of a model on a finite Cayley tree, where wave function moments exhibit multifractal scaling with $N$ in the limit of large $N$. The multifractality spectrum depends on disorder strength and on the position of the lattice, as we show both analytically and numerically [4, 5]. Finally, I will briefly discuss connections between the (de-)localization on RRG and the many-body localization [1, 6].

[1] K. S. Tikhonov and A. D. Mirlin, Phys. Rev. B 99, 024202 (2019)
[2] K. S. Tikhonov, A. D. Mirlin, and M.A. Skvortsov, Phys. Rev. B 94, 220203 (2016)
[3] K. S. Tikhonov and A. D. Mirlin, Phys. Rev. B 99, 214202 (2019)
[4] K. S. Tikhonov and A. D. Mirlin, Phys. Rev B 94, 184203 (2016)
[5] M. Sonner, K. S. Tikhonov and A. D. Mirlin, Phys. Rev. B 96, 214204 (2017)
[6] K. S. Tikhonov and A. D. Mirlin, Phys. Rev. B 97, 214205 (2018) and in preparation