# Diffusive estimates for random walk under annealed polynomial growth

We show that on a random infinite graph G of polynomial growth where simple random walk is stationary, it is diffusive along a subsequence, i.e., the second moment of the distance from the starting point grows linearly in time. This extends a result of Kesten that applied to the extrinsic metric on subgraphs of the lattice Zd, and answers a question due to Benjamini, Duminil-Copin, Kozma and Yadin. We also show that, in general, passing to a subsequence is necessary. As a consequence, we deduce that harmonic functions of sublinear growth on such graphs G are constants. Our proof combines embeddings with the mass transport principle. Based on joint work with James Lee and Yuval Peres.