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.
"Loop-erased random walk and Poisson kernel on planar graphs." Ann. Probab. 39 (4) 1243 - 1285, July 2011. https://doi.org/10.1214/10-AOP579