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

TL;DR

This paper introduces a statistical model of random 3-regular graphs using Ollivier curvature, demonstrating that in the limit, these graphs converge to a one-dimensional circle, providing insights into emergent geometry in quantum gravity.

Contribution

The paper shows that random 3-regular graphs with certain constraints converge to a circle, offering a mathematically rigorous and computationally accessible model of emergent one-dimensional geometry.

Findings

Hausdorff and spectral dimensions approach 1

Convergence to a circle in the Gromov-Hausdorff sense

Evidence of a second-order phase transition

Abstract

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 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.