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We consider from the localization perspective the new critical phenomenadiscovered recently for perturbed random regular graphs (RRG) and constrainedErd\H{o}s-R\'enyi networks (CERN) \cite{crit2}. At some critical value of thechemical potential of 3-cycles, $\mu$, the network decays into the maximallypossible number of almost full subgraphs, and the spectrum of the Laplacianmatrix acquires the two-zonal structure with a large gap. We find that theLaplacian eigenvalue statistics corresponds to delocalized states in one zone,and to the localized states in the second one. We interpret this behavior interms of the many-body localization problem where the structure of the Fockspace of some interacting many-body system is approximated by the RGG and/or bythe CERN. We associate 3-cycles in RRGs and CERNs as resonant triples in theFock space. We show that the scenario of the "localization without disorder",discussed previously in physical space, can be realized in the Fock space aswell. We argue that it is natural to identify clusters in a RRG with particlesin a many-body system above the phase transition. We discuss the controversialissue of an additional phase transition between ergodic and non-ergodic regimesin the delocalized phase in the Fock space and find a strong "memorydependence" of the states in the delocalized phase, thus advocating existenceof non-ergodic delocalized states.

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