Loop-Erased Random Walk and Poisson Kernel on Planar Graphs

Details Loop-Erased Random Walk and Poisson Kernel on Planar Graphs Loop-Erased Random Walk and Poisson Kernel on Planar Graphs

2008-09-16

arxiv.org/abs/0809.2643v1

Mathematics

Probability

Scientific paper

Lawler, Schramm and Werner showed that the scaling limit of the loop-erased random walk on the square lattice is SLE(2). We consider scaling limits of the loop-erasure of random walks on other planar graphs (graphs embedded into the complex plane 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 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 the square lattice. 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 SLE(2).

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