Emergence of the Circle in a Statistical Model of Random Cubic Graphs

Abstract

We consider a formal discretisation of Euclidean quantum gravity defined by a statistical model of random 3regular graphs and making using of the Ollivier curvature, a coarse analogue of the Ricci curvature. Numerical analysis shows that the Hausdorff and spectral dimensions of the model approach 1 in the joint classical-thermodynamic limit and we argue that the scaling limit of the model is the circle of radius r, S 1 r . Given mild kinematic constraints, these claims can be proven with full mathematical rigour: speaking precisely, it may be shown that for 3-regular graphs of girth at least 4, any sequence of action minimising configurations converges in the sense of Gromov-Hausdorff to S 1 r . We also present strong evidence for the existence of a second-order phase transition through an analysis of finite size effects. This-essentially solvable-toy model of emergent one-dimensional geometry is meant as a controllable paradigm for the nonperturbative definition of random flat surfaces.

Interacting Thermofield Doubles and Critical Behavior in Random Regular Graphs

“…The number of nodes involved into the closed ribbon depends on µ 4 . Our finding contradicts the claim in [23][24][25] about the order of the phase transition for d > 3 and about the structure of emerging phase.…”

“…We reproduce the result of [24] for d = 3 that at µ 4 > µ cRRG 4 the single composite bipartite closed ribbon emerges. However for d > 3 we clearly demonstrate that the clustering into hypercubes exists contrary to the claim in [23][24][25].…”

“…The authors of [23][24][25] claimed that there is the phase transition at some µ cRRG 4 which has the different nature for RRG and cRRG. The phase transition in cRRG was assumed to be the second order transition without formation of clusters.…”

“…The dimension of emerging Einstein-Hilbert action D is defined by the degree d in RRG and µ 4 upon proper rescaling plays a role of the gravitational coupling. It was claimed in [23,25] that such model undergoes the second order phase transition into the dense phase at some µ c 4 without a cluster formation. The d = 3 example has been investigated in [24] and the emerging S 1 geometry has been identified.…”

Quantum Flatness in Two-Dimensional CDT Quantum Gravity

“…Our philosophy here is different from that of "combinatorial quantum gravity"[33,34,35,36], which investigates the emergence of geometry from specific ensembles of random graphs, with the ambition of having its Lorentzian structure emerge alongside, something not yet achieved by any model as far as we know. In this approach, one uses the Ollivier-Ricci curvature in its original form for small values of δ, and has recently also compared it to another discrete notion of so-called Forman-Ricci curvature[37].…”