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 $d<\infty$. We prove that there exists a constant $C_d$ such that the continuous-time random walk displacement and log-volume growth satisfy $$\mathbb{E}_x \mathrm{dist}(x,X_t)^2 \le t \exp\left[C_d \sqrt{\log t \log\log t}\right],$$ $$\log \mathrm{Vol}(B(x,r)) \le \exp\left[C_d \sqrt{\log r \log\log r}\right],$$ for every $x\in V$ and all $r,t \ge e^e$.