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June 2014 Convergence of loop erased random walks on a planar graph to a chordal SLE(2) curve
Hiroyuki Suzuki
Kodai Math. J. 37(2): 303-329 (June 2014). DOI: 10.2996/kmj/1404393889

Abstract

In this paper we consider the 'natural' random walk on a planar graph and scale it by a small positive number δ. Given a simply connected domain D and its two boundary points a and b, we start the scaled walk at a vertex of the graph nearby a and condition it on its exiting D through a vertex nearby b, and prove that the loop erasure of the conditioned walk converges, as δ ↓ 0, to the chordal SLE2 that connects a and b in D, provided that an invariance principle is valid for both the random walk and the dual walk of it. Our result is an extension of one due to Dapeng Zhan [12] where the problem is considered on the square lattice. A convergence to the radial SLE2 has been obtained by Lawler, Schramm and Werner [3] for the square and triangular lattices and by Yadin and Yehudayoff [10] for a wide class of planar graphs. Our proof, though an adaptation of that of [3] and [10], involves some new ingredients that arise from two sources: one for dealing with a martingale observable that is different from that used in [3] and [10] and the other for estimating the harmonic measures of the random walk started at a boundary point of a domain.

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Hiroyuki Suzuki. "Convergence of loop erased random walks on a planar graph to a chordal SLE(2) curve." Kodai Math. J. 37 (2) 303 - 329, June 2014. https://doi.org/10.2996/kmj/1404393889

Information

Published: June 2014
First available in Project Euclid: 3 July 2014

zbMATH: 1306.60117
MathSciNet: MR3229078
Digital Object Identifier: 10.2996/kmj/1404393889

Rights: Copyright © 2014 Tokyo Institute of Technology, Department of Mathematics

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Vol.37 • No. 2 • June 2014
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