Interacting thermofield doubles and critical behavior in random regular graphs

We discuss numerically the nonperturbative effects in exponential random graphs which are analogue of eigenvalue instantons matrix models. The phase structure with chemical potential for ${C}_{4}$ ${\ensuremath{\mu}}_{4}$ and degree preserving constraint is clarified. first order transition at critical value ${\ensuremath{\mu}}_{4}^{\mathrm{RRG}}$ into bipartite a formation fixed number clusters found ensemble regular (RRG). consider similar mean field version combinatorial quantum gravity based Ollivier graph curvature RRG supplemented hard-core show that ${\ensuremath{\mu}}_{4}^{\mathrm{CRRG}}$ emerging depend on vertex $d$ RRG. For $d=3$ closed ribbon emerges ${\ensuremath{\mu}}_{4}>{\ensuremath{\mu}}_{4}^{\mathrm{CRRG}}$ while $d>3$ isolated or weakly interacting hypercubes gets emerged clear-cut hysteresis. If additional connectedness condition imposed identified as chain coupled hypercubes. Since ground state hypercube thermofield double we suggest dual holographic picture involves multiboundary wormholes. Treating model Hilbert space many-body system patterns fragmentation transition. also briefly comment possible relation to problem interpretation partial deconfinement gauge theories.