Pointwise subexponential growth and near-diffusive displacement on bounded-degree graphs with non-negative Ollivier--Ricci curvature

Let $G=(V,E)$ be a possibly infinite, locally finite graph with non-negative Ollivier–Ricci curvature and degrees bounded by $d<\mathrm{\infty }$. We prove that there exists a constant $C_d$ such that the continuous-time random walk displacement and log-volume growth satisfy $${\mathbb{E}}_x\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,X_t{)}^2\le t\exp \left[C_d\sqrt{\log t\log \log t}\right],$$ $$\log \mathrm{V}\mathrm{o}\mathrm{l}(B(x,r))\le \exp \left[C_d\sqrt{\log r\log \log r}\right],$$ for every $x\in V$ and all $r,t\ge e^e$.