Bounded-degree graphs of non-negative Ollivier-Ricci curvature have subexponential growth and diffusive random walk

TL;DR

This paper demonstrates that graphs with bounded degree and non-negative Ollivier-Ricci curvature exhibit subexponential volume growth and diffusive random walks, extending previous results and applying to infinite transitive and unimodular random graphs.

Contribution

It proves subexponential growth and diffusive behavior for bounded-degree graphs with non-negative Ollivier-Ricci curvature, strengthening prior results and extending applicability.

Findings

Graphs with non-negative Ollivier-Ricci curvature have subexponential volume growth.

Random walks on these graphs are diffusive with displacement n^{1+o(1)}.

Results apply to infinite transitive and unimodular random graphs.

Abstract

We study the geometric properties of graphs with non-negative Ollivier-Ricci curvature, a discrete analogue of non-negative Ricci curvature in Riemannian geometry. We prove that for each $d<\infty$ there exists a constant $C_d$ such that if $G=(V,E)$ is a finite graph with non-negative Ollivier-Ricci curvature and with degrees bounded by $d$ then the average log-volume growth and random walk displacement satisfy \[ \frac{1}{|V|} \sum_{x\in V} \log \#B(x,r) \leq \exp\left[C_d \sqrt{\log r}\right] = r^{o(1)} \] and \[ \frac{1}{|V|} \sum_{x\in V} \mathbf{E}_x [d(X_0,X_n)^2] \leq n \exp\left[C_d \sqrt{\log n}\right] = n^{1+o(1)} \] for every $n,r\geq 2$. This significantly strengthens a result of Salez (GAFA 2022), who proved that the average displacement of the random walk is $o(n)$ and deduced that non-negatively curved graphs of bounded degree cannot be expanders. Our results also apply to infinite transitive graphs and, more generally, to bounded-degree unimodular random rooted graphs of non-negative Ollivier-Ricci curvature.