Interacting Thermofield Doubles and Critical Behavior in Random Regular Graphs

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 µ 4 and degree preserving constraint is clarified. The first order phase transition at critical value of chemical potential for 4-cycles µ RRG 4

Convergence of Combinatorial Gravity

“…The basic idea is that A MF promotes the formation of squares; at fixed N there is nothing to oppose this tendency in A MF and in this sense it is not surprising that there is a first-order transition to a phase characterised by separated (if we allow configurations to be disconnected) or weakly interacting (if we demand that configurations remain connected) hypercubes, since the latter maximise the number of squares per edge at given degree. Indeed in [47] we observed separated hypercubes, which we somewhat inappropriately referred to as baby universes: compare figure 5 of [47] with figure 4 of [35]. On the other hand, the term defined as a sum over P (G) acts in opposition to A MF , once the degree of clustering reaches a certain point.…”

“…Moreover combinatorial quantum gravity appears to have a continuous phase transition between a phase of random graphs and a phase evincing emergent geometric structure. Gorsky and Valba [35] have recently examined an approximation to combinatorial quantum gravity-studied briefly in [47]-and showed that the resulting model displays quite a different phase structure; indeed in this approximate model there is a first-order transition between a random graph phase and a phase characterised by weakly interacting hypercubes. Other than combinatorial quantum gravity, Loll and Klitgaard have recently advocated the use of a quantum Ricci curvature, inspired by the Ollivier curvature, as a quasilocal observable in nonperturbative quantum gravity [49,50,51]; Gorard has also advocated the use of the Ollivier curvature in the context of Wolfram's recent work on graphs [34,92].…”

“…As mentioned previously, Gorsky and Valba [35] have recently studied an approximation of the nonconvergent model of combinatorial quantum gravity introduced in [47,81] and have found quite different results for the phase structure and nature of the observed phase transition. Moreover, Gorsky and Valba claim that the model studied in [35] is the same as the model we consider in [47,48]. It is worth stressing that this apparent conflict is in fact the result of a misunderstanding: Gorsky and Valba study a model based on the action:…”

“…In [47], we referred to this as a mean field approximation because the term A MF depends only on global quantities. This mean field approximation is also equivalent to ignoring the subscript + in the expression 26 (compare with equation 2.6 of [35]). The basic idea is that A MF promotes the formation of squares; at fixed N there is nothing to oppose this tendency in A MF and in this sense it is not surprising that there is a first-order transition to a phase characterised by separated (if we allow configurations to be disconnected) or weakly interacting (if we demand that configurations remain connected) hypercubes, since the latter maximise the number of squares per edge at given degree.…”