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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 u003e 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 u003e 0}) such that $$mathbb{E} left[mathrm{dist}_G(X_0, X_{t_n})^2 mid X_0 = rho, (G,rho)right]leq C t_nqquad forall n geq 1,.$$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.

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