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

Marek Biskup, Jian Ding, Subhajit Goswami

Published 2016-11-11Version 1

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 $N^{o(1)}$.