429 research outputs found
Loop-erased random walk and Poisson kernel on planar graphs
Lawler, Schramm and Werner showed that the scaling limit of the loop-erased
random walk on is . We consider scaling limits
of the loop-erasure of random walks on other planar graphs (graphs embedded
into 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 . 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 . 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 .Comment: Published in at http://dx.doi.org/10.1214/10-AOP579 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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