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

Abstract

Let (G, ρ) be a stationary random graph, and use B G ρ (r) to denote the ball of radius r about ρ in G. Suppose that (G, ρ) has annealed polynomial growth, in the sense that E[|B G ρ (r)|] O(r k ) for some k > 0 and every r 1.Then there is an infinite sequence of times {t n } at which the random walk {X t } on (G, ρ) is at most diffusive: Almost surely (over the choice of (G, ρ)), there is a number C > 0 such thatThis result is new even in the case when G is a stationary random subgraph of 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.

Uniqueness of the infinite tree in low-dimensional random forests

“…The four-dimensional case is more delicate since this dimension is critical for Z d to have the infinite intersection property, with each dyadic scale only contributing O(1) intersections in expectation. We believe that it should be possible to extend Theorem 1.5 to unimodular random rooted graphs whose balls have volume O(n 4 ) using the methods of Ganguly, Lee, and Peres [21], who proved that any unimodular random rooted graph of polynomial volume growth satisfies a diffusive estimate at infinitely many scales.…”