From Anderson localization on random regular graphs to many-body localization

The article reviews the physics of Anderson localization on random regular graphs (RRG) and its connections to many-body localization (MBL) in disordered interacting systems. Properties of eigenstate and energy level correlations in delocalized and localized phases, as well at criticality, are discussed. In the many-body part, models with short-range and power-law interactions are considered, as well as the quantum-dot model representing the limit of the “most long-range” interaction. Central themes – which are common to the RRG and MBL problems – include ergodicity of the delocalized phase, localized character of the critical point, strong finite-size effects, and fractal scaling of eigenstate correlations in the localized phase. •Anderson transition from ergodicity to localization on random regular graphs (RRG).•Analytical, pool method, and exact-diagonalization study of correlations on RRG.•Many-body localization (MBL) to ergodicity transition: quantum dots and spin chains.•Dynamical eigenstate correlation functions in RRG and MBL problems.•Anderson localization on RRG as a toy model for MBL.