Open Access
Translator Disclaimer
July 2011 Loop-erased random walk and Poisson kernel on planar graphs
Ariel Yadin, Amir Yehudayoff
Ann. Probab. 39(4): 1243-1285 (July 2011). DOI: 10.1214/10-AOP579

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.

Citation

Download Citation

Ariel Yadin. Amir Yehudayoff. "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

Information

Published: July 2011
First available in Project Euclid: 5 August 2011

zbMATH: 1234.60036
MathSciNet: MR2857239
Digital Object Identifier: 10.1214/10-AOP579

Subjects:
Primary: 60F17 , 60J99 , 60K35

Keywords: Loop-erased random walk , Planar graphs , Poisson kernel , Schramm–Loewner evolution

Rights: Copyright © 2011 Institute of Mathematical Statistics

JOURNAL ARTICLE
43 PAGES


SHARE
Vol.39 • No. 4 • July 2011
Back to Top