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 et al. (Ann Probab 43(5):2332–2373, 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.