Diffusive estimates for random walks on stationary random graphs of polynomial growth

Abstract: Let $(G,\rho)$ be a stationary random graph, and use $B^G_{\rho}(r)$ to denote the ball of radius $r$ about $\rho$ in $G$. Suppose that $(G,\rho)$ has annealed polynomial growth, in the sense that $\mathbb{E}[|B^G_{\rho}(r)|] \leq O(r^k)$ for some $k > 0$ and every $r \geq 1$. Then there is an infinite sequence of times ${t_n}$ at which the random walk ${X_t}$ on $(G,\rho)$ is at most diffusive: Almost surely (over the choice of $(G,\rho)$), there is a number $C > 0$ such that [ \mathbb{E} \left[\mathrm{dist}G(X_0, X{t_n})^2 \mid X_0 = \rho, (G,\rho)\right]\leq C t_n\qquad \forall n \geq 1\,. ] This result is new even in the case when $G$ is a stationary random subgraph of $\mathbb{Z}^d$. Combined with the work of Benjamini, Duminil-Copin, Kozma, and Yadin (2015), it implies that $G$ almost surely does not admit a non-constant harmonic function of sublinear growth. To complement this, we argue that passing to a subsequence of times ${t_n}$ is necessary, as there are stationary random graphs of (almost sure) polynomial growth where the random walk is almost surely superdiffusive at an infinite subset of times.