Convergence of Combinatorial Gravity

TL;DR

This paper introduces a graph-based regularisation of Euclidean Einstein gravity using Ollivier curvature, enabling the approximation of manifold geometries and fields through dense random geometric graphs, paving the way for a combinatorial quantum gravity approach.

Contribution

It defines a discrete Einstein-Hilbert action on graphs that converges to the continuum on dense graphs, utilizing Ollivier curvature, and extends this framework to Klein-Gordon fields.

Findings

Discrete Einstein-Hilbert action converges to the manifold version on dense graphs.

Method uses Ollivier curvature from optimal transport theory.

Framework supports a combinatorial approach to quantum gravity.

Abstract

We present a new regularisation of Euclidean Einstein gravity in terms of (sequences of) graphs. In particular, we define a discrete Einstein-Hilbert action that converges to its manifold counterpart on sufficiently dense random geometric graphs (more generally on any sequence of graphs that converges to the manifold in the sense of Gromov-Hausdorff). Our construction relies crucially on the Ollivier curvature of optimal transport theory. Our methods also allow us to define an analogous discrete action for Klein-Gordon fields. These results may be taken as the basis for a combinatorial approach to quantum gravity where we seek to generate graphs that approximate manifolds as metric-measure structures.