Abstract
We discuss numerically the non-perturbative effects in exponential random graphs which are analogue of eigenvalue instantons in matrix models. The phase structure of exponential random graphs with chemical potential for 4-cycles and degree preserving constraint is clarified. The first order phase transition at critical value of chemical potential for 4-cycles into bipartite phase with a formation of fixed number of bipartite clusters is found for ensemble of random regular graphs (RRG). We consider the similar phase transition in combinatorial quantum gravity based of the Ollivier graph curvature for RRG supplemented with hard-core constraint and show that a order of a phase transition and the structure of emerging phase depend on a vertex degree d in RRG. For d = 3 the bipartite closed ribbon emerges at bipartite phase while for d > 3 the ensemble of isolated or weakly interacting hypercubes supplemented with the bipartite closed ribbon gets emerged at the first order phase transition with a clear-cut hysteresis. If the additional connectedness condition is imposed the bipartite phase gets identified as the closed chain of weakly coupled hypercubes. Since the ground state of isolated hypercube is the thermofield double (TFD) we suggest that the dual holographic picture involves multiboundary wormholes. Treating RRG as a model of a Hilbert space for a interacting many-body system we discuss the patterns of the Hilbert space fragmentation at the phase transition. We also briefly comment on a possible relation of the found phase transition to the problem of holographic interpretation of a partial deconfinement transition in the gauge theories.
Highlights
Matrix models play a prominent role in a description of chaotic systems with large number degrees of freedom
It turns out that degree conservation constraint amounts to the first order phase transition with formation of the multiple weakly interacting clusters which has been demonstrated for constrained ER and random regular graphs (RRG) with Hamiltonian involving μ3 ≠ 0 [11]
They are similar to the eigenvalue instantons in matrix models the degree conservation constraint corresponds to the account of the non-singlet sector in matrix model context
Summary
Matrix models play a prominent role in a description of chaotic systems with large number degrees of freedom. It turns out that degree conservation constraint amounts to the first order phase transition with formation of the multiple weakly interacting clusters which has been demonstrated for constrained ER and random regular graphs (RRG) with Hamiltonian involving μ3 ≠ 0 [11]. In the ER model with local constraint and RRG the formation of network cluster corresponds exactly to the single eigenvalue instanton. The dimension of emerging Einstein-Hilbert action D is defined by the degree d in RRG and μ4 upon proper rescaling plays a role of the gravitational coupling It was claimed in [21,25] that such model undergoes the second order phase transition into the dense phase at some μc without a cluster formation.
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