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

TL;DR

This paper derives precise asymptotic behaviors for potential kernels and Green's functions of two-dimensional random walks in random conductance environments, including percolation clusters and dynamic models, with applications to interface models.

Contribution

It provides new asymptotic results for potential kernels and Green's functions in complex random environments, extending previous work to degenerate and dynamic conductance models.

Findings

Asymptotics established for potential kernel and Green's function.

Results apply to i.i.d. supercritical percolation clusters.

Scaling limit derived for variances in the Ginzburg-Landau model.

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 $\nabla \phi$-interface model.