Emergence of the circle in a statistical model of random cubic graphs

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, Sr1. 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 Sr1. 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.