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

TL;DR

Bounding-degree graphs with non-negative Ollivier-Ricci curvature exhibit subexponential volume growth and near-diffusive random-walk displacement. The authors develop a grand Poisson cell decomposition that ties random-walk entropy to cell sizes, derive a functional inequality balancing isoperimetric and entropy terms, and bootstrap to obtain explicit entropy and growth bounds of the form H_n ≤ d^3 exp(6 sqrt{2 log(AC) log n}). These results extend to infinite transitive and unimodular random rooted graphs and improve prior displacement bounds, offering quantitative control over geometry and mixing. The paper further discusses conjectures suggesting polynomial growth and a Cheeger–Gromoll-type structure for transitive graphs, and outlines potential extensions to Bakry–Emery curvature. Overall, the work provides a quantitative bridge between local non-negative Ollivier-Ricci curvature and global subexponential growth and diffusion behavior in bounded-degree 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.

Theorems & Definitions (53)

• Theorem 1.1
• Remark 1.2: The role of laziness
• Remark 1.3: Other notions of curvature
• Remark 1.4: Weighted graphs
• Theorem 1.5
• Theorem 2.1
• Theorem 2.2
• Proposition 2.3
• proof : Proof of \ref{['prop:Lipschitz_concentration']}
• Remark 2.4