Return probability and recurrence for the random walk driven by two-dimensional Gaussian free field

Abstract

Given any $\gamma >0$ and for $\eta =\{{\eta }_v{\}}_{v\in {\mathbb{Z}}^2}$ denoting a sample of the two-dimensional discrete Gaussian free field on ${\mathbb{Z}}^2$ pinned at the origin, we consider the random walk on~${\mathbb{Z}}^2$ among random conductances where the conductance of edge $(u,v)$ is given by ${\mathrm{e}}^{\gamma ({\eta }_u+{\eta }_v)}$. We show that, for almost every~$\eta$, this random walk is recurrent and that, with probability tending to~1 as $T\to \infty$, the return probability at time~$2T$ decays as $T^{-1+o(1)}$. In addition, we prove a version of subdiffusive behavior by showing that the expected exit time from a ball of radius~$N$ scales as $N^{\psi (\gamma )+o(1)}$ with $\psi (\gamma )>2$ for all~$\gamma >0$. Our results rely on delicate control of the effective resistance for this random network. In particular, we show that the effective resistance between two vertices at Euclidean distance~$N$ behaves as~$N^{o(1)}$.

58 pages, 10 figures. The current version has been accepted for publication in Communications in Mathematical Physics

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Cited by in corpus (1)

• First passage percolation with long-range correlations and applications to random Schrödinger operators