Loop-erased random walk and Poisson kernel on planar graphs

TL;DR

This paper proves that on certain planar graphs where random walks converge to planar Brownian motion, the loop-erased random walk converges to SLE_2, extending known results beyond the square lattice.

Contribution

It demonstrates that the scaling limit of loop-erased random walks on planar graphs with Brownian motion limits is SLE_2, and shows the discrete Poisson kernel approximates the continuous one.

Findings

Loop-erased random walk on these graphs converges to SLE_2.

Discrete Poisson kernel approximates the continuous Poisson kernel.

Results apply to super-critical percolation clusters.

Abstract

Lawler, Schramm and Werner showed that the scaling limit of the loop-erased random walk on $\mathbb{Z}^2$ is $\mathrm{SLE}_2$. We consider scaling limits of the loop-erasure of random walks on other planar graphs (graphs embedded into $\mathbb{C}$ so that edges do not cross one another). We show that if the scaling limit of the random walk is planar Brownian motion, then the scaling limit of its loop-erasure is $\mathrm{SLE}_2$. Our main contribution is showing that for such graphs, the discrete Poisson kernel can be approximated by the continuous one. One example is the infinite component of super-critical percolation on $\mathbb{Z}^2$. Berger and Biskup showed that the scaling limit of the random walk on this graph is planar Brownian motion. Our results imply that the scaling limit of the loop-erased random walk on the super-critical percolation cluster is $\mathrm{SLE}_2$.