Green kernel asymptotics for two-dimensional random walks under random conductances

Abstract

We consider random walks among random conductances on Z 2 and establish precise asymptotics for the associated potential kernel and the Green function of the walk killed upon exiting balls. The result is proven under a general set of assumptions, where examples include uniformly elliptic conductances, random walks on supercritical percolation clusters and ergodic degenerate conductances satisfying a moment condition. We also provide a similar result for the time-dynamic random conductance model. As an application we present a scaling limit for the variances in the Ginzburg-Landau ∇φ-interface model. CONTENTS1. Introduction 1 2. Green kernel asymptotics in the two-dimensional static RCM 6 2.1. Assumptions 6 2.2. Examples 8 2.3. Proof of Theorem 1.4 9 2.4. Proof of Theorem 1.5 10 3. Green kernel asymptotics in the two-dimensional dynamic RCM 12 3.1. Setting and results 12 3.2. Application to stochastic interface models 14 References 15

Quantitative homogenization of the parabolic and elliptic Green's functions on percolation clusters

“…Results on the elliptic Green's function usually follow from the ones established on the parabolic Green's function, by an application of the formula (1.8) in dimension larger than 3. In dimension 2 the situation is different and requires separate considerations; in [8], Andres, Deuschel and Slowik characterize the asymptotics of the Green's function associated to the random walk killed upon exiting a ball under general assumptions on the environment.…”