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

Abstract

We consider random walks among random conductances on ${\mathbb{Z}}^2$ and establish precise asymptotics for the associated potential kernel and the Green's function of the walk killed upon exiting balls. The result is proven for random walks on i.i.d. supercritical percolation clusters among 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 $\nabla \varphi$-interface model.

17 pages; accepted version

References in corpus (4)

• Random walks on supercritical percolation clusters
• Invariance principle for the random conductance model with unbounded conductances
• Anomalous heat-kernel decay for random walk among bounded random conductances
• Limit theory for random walks in degenerate time-dependent random environments

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