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

TL;DR

This paper studies a random walk on a two-dimensional lattice with conductances driven by a Gaussian free field, showing it is recurrent with subdiffusive behavior and analyzing the effective resistance in the network.

Contribution

It establishes recurrence, return probability decay, and subdiffusive scaling for the random walk driven by the Gaussian free field, with detailed resistance estimates.

Findings

Random walk is recurrent for almost every Gaussian free field sample.

Return probability at time 2T decays as T^{-1+o(1)}.

Expected exit time from a ball scales as N^{9(\u03b3)+o(1)} with 9(b3)>2.

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~$N^{o(1)}$.