Diffusive estimates for random walks on stationary random graphs of polynomial growth
Geometric and Functional Analysis(2017)
摘要
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 that 𝔼[dist_G(X_0, X_t_n)^2 | X_0 = ρ, (G,ρ)]≤ C t_n ∀ n ≥ 1 . This result is new even in the case when G is a stationary random subgraph of 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.
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关键词
stationary random graphs,random walks,diffusive estimates
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