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

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 $d<\mathrm{\infty }$ 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 $$\frac{1}{\mathrm{\mid }V\mathrm{\mid }}\sum _{x\in V}\log \mathrm{\#}B(x,r)\le \exp \left[C_d\sqrt{\log r}\right]=r^{o(1)}$$ and $$\frac{1}{\mathrm{\mid }V\mathrm{\mid }}\sum _{x\in V}{\mathbf{E}}_x[d(X_0,X_n{)}^2]\le n\exp \left[C_d\sqrt{\log n}\right]=n^{1+o(1)}$$ for every $n,r\ge 2$. 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.