The Annals of Probability

Random walks and harmonic functions on infinite planar graphs using square tilings

Itai Benjamini and Oded Schramm

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Abstract

We study a wide class of transient planar graphs, through a geometric model given by a square tiling of a cylinder. For many graphs, the geometric boundary of the tiling is a circle and is easy to describe in general. The simple random walk on the graph converges (with probability 1) to a point in the geometric boundary. We obtain information on the harmonic measure and estimates on the rate of convergence. This allows us to extend results we previously proved for triangulations of a disk.

Article information

Source
Ann. Probab., Volume 24, Number 3 (1996), 1219-1238.

Dates
First available in Project Euclid: 9 October 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1065725179

Digital Object Identifier
doi:10.1214/aop/1065725179

Mathematical Reviews number (MathSciNet)
MR1411492

Zentralblatt MATH identifier
0862.60053

Subjects
Primary: 60J15 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 52C20: Tilings in $2$ dimensions [See also 05B45, 51M20]

Keywords
Planar graphs random walks harmonic measure Dirichlet problem

Citation

Benjamini, Itai; Schramm, Oded. Random walks and harmonic functions on infinite planar graphs using square tilings. Ann. Probab. 24 (1996), no. 3, 1219--1238. doi:10.1214/aop/1065725179. https://projecteuclid.org/euclid.aop/1065725179


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