Loop-erased random walk and Poisson kernel on planar graphs

Abstract

Lawler, Schramm and Werner showed that the scaling limit of the loop-erased random walk on ℤ² is SLE₂. We consider scaling limits of the loop-erasure of random walks on other planar graphs (graphs embedded into 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 SLE₂. 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 ℤ². 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 SLE₂.