  Anderson localization on random regular graphs (RRG) serves as a toy-model of
many-body localization (MBL). We explore the transition for ergodicity to
localization on RRG with large connectivity $m$. In the analytical part, we
focus on the inverse participation ratio of eigenstates and identify several
regimes on the ergodic side of the transition -- self-consistent golden rule,
golden rule, precritical, and critical -- that the system consecutively goes
through when the disorder increases towards the point $W_c$ of the localization
transition. We also perform exact-diagonalization numerics as well as a
population-dynamics analysis that combines analytical and numerical techniques.
Results of all the approaches are in excellent mutual agreement. We further
explore the evolution from ergodicity to localization in two models of
Fock-space MBL: fermionic and spin quantum dots. Large-connectivity RRG models
serve as an approximation for this class of many-body problems; one of our
central goals is to better understand the status of this approximation. Our
numerical simulations support the conjecture that, in the limit of a large
system, there is a sharp ergodicity-to-localization transition in the
quantum-dot models. Furthermore, our results are consistent the RRG-like
scaling of the critical disorder, $W_c \sim m \ln m$. While in the golden-rule
range the behavior of quantum-dot models is in agreement with analytical
predictions based on RRG model, substantial deviations occur in the
pre-critical regime. Our results indicate that the pre-critical and critical
behavior (as well as the numerical coefficient in the formula for $W_c$) in the
quantum-dot models may be different from what one would expect from the
RRG-like approximation.
