Many-body localization and new critical phenomena in regular random graphs and constrained Erd\H{o}s-Renyi networks

We consider from the localization perspective the new critical phenomena discovered recently for perturbed random regular graphs (RRG) and constrained Erd\H{o}s-R\'enyi networks (CERN) \cite{crit2}. At some critical value of the chemical potential of 3-cycles, $\mu$, the network decays into the maximally possible number of almost full subgraphs, and the spectrum of the Laplacian matrix acquires the two-zonal structure with a large gap. We find that the Laplacian eigenvalue statistics corresponds to delocalized states in one zone, and to the localized states in the second one. We interpret this behavior in terms of the many-body localization problem where the structure of the Fock space of some interacting many-body system is approximated by the RGG and/or by the CERN. We associate 3-cycles in RRGs and CERNs as resonant triples in the Fock space. We show that the scenario of the "localization without disorder", discussed previously in physical space, can be realized in the Fock space as well. We argue that it is natural to identify clusters in a RRG with particles in a many-body system above the phase transition. We discuss the controversial issue of an additional phase transition between ergodic and non-ergodic regimes in the delocalized phase in the Fock space and find a strong "memory dependence" of the states in the delocalized phase, thus advocating existence of non-ergodic delocalized states.