Emergent geometry and discrete curvature in random graphs

Abstract

In this thesis, we explore various themes relating to the emergence of geometry in (ensembles of random) graphs in connection with the notion of Ollivier curvature, a synthetic generalisation of the manifold Ricci curvature. In particular we study the problem of computing the Ollivier curvature in classes of graphs, presenting the most general valid explicit expressions available. These exact expressions greatly facilitate the analytic and numerical study of a model of random graphs dubbed combinatorial quantum gravity by its initiator Trugenberger [282], essentially defined by a formal discretisation of the Einstein-Hilbert action. Restricting to a configuration space of cubic graphs allows us to show both analytically and numerically that there is a geometric phase consisting of a discrete cylinder/M¨obius strip. The scaling limit—formally a Gromov-Hausdorff limit—of these configurations turns out to be the circle, while numerical evidence suggests that the transition to this geometric phase is continuous. The critical temperature appears to be asymptotically nonfinite. Higher degree results are less conclusive and the phase transition appears to become first-order, but there are good analytical and numerical indications that configurations retain a nearly geometric phase. We also develop a geometric convergence theory for the Ollivier curvature in discrete graphs that approximate (compact) Riemannian manifolds in the sense of Gromov-Hausdorff and find a convergent discrete Einstein-Hilbert action. Our results thus present some of the best concrete models of geometrogenesis in discrete structures and show that the Ollivier curvature can be used as the basis for a regularisation of classical Euclidean spacetime in terms of networks.