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

Given any $γ>0$ and for $η=\{η_v\}_{v\in \mathbb Z^2}$ 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 $\mathrm{e}^{γ(η_u + η_v)}$. We show that, for almost every~$η$, 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 $N^{ψ(γ)+o(1)}$ with $ψ(γ)>2$ for all~$γ>0$. 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~$N^{o(1)}$.

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