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Anderson localization on random regular graphs (RRG) serves as a toy-model ofmany-body localization (MBL). We explore the transition for ergodicity tolocalization on RRG with large connectivity $m$. In the analytical part, wefocus on the inverse participation ratio of eigenstates and identify severalregimes on the ergodic side of the transition -- self-consistent golden rule,golden rule, precritical, and critical -- that the system consecutively goesthrough when the disorder increases towards the point $W_c$ of the localizationtransition. We also perform exact-diagonalization numerics as well as apopulation-dynamics analysis that combines analytical and numerical techniques.Results of all the approaches are in excellent mutual agreement. We furtherexplore the evolution from ergodicity to localization in two models ofFock-space MBL: fermionic and spin quantum dots. Large-connectivity RRG modelsserve as an approximation for this class of many-body problems; one of ourcentral goals is to better understand the status of this approximation. Ournumerical simulations support the conjecture that, in the limit of a largesystem, there is a sharp ergodicity-to-localization transition in thequantum-dot models. Furthermore, our results are consistent the RRG-likescaling of the critical disorder, $W_c \sim m \ln m$. While in the golden-rulerange the behavior of quantum-dot models is in agreement with analyticalpredictions based on RRG model, substantial deviations occur in thepre-critical regime. Our results indicate that the pre-critical and criticalbehavior (as well as the numerical coefficient in the formula for $W_c$) in thequantum-dot models may be different from what one would expect from theRRG-like approximation.

Ergodicity-to-localization transition on random regular graphs with large connectivity and in many-body quantum dots