Emergence of the Circle in a Statistical Model of Random Cubic Graphs [article]

Christy Kelly, Carlo Trugenberger, Fabio Biancalana
2021 arXiv   pre-print
We consider a formal discretisation of Euclidean quantum gravity defined by a statistical model of random 3-regular 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
more » ... thematical 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.
arXiv:2008.11779v3 fatcat:4sopddxir5f45kr7aar33cix3a