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Random Walks and Harmonic Functions on Infinite Planar Graphs Using Square Tilings

Itai Benjamini and Oded Schramm
The Annals of Probability
Vol. 24, No. 3 (Jul., 1996), pp. 1219-1238
Stable URL: http://www.jstor.org/stable/2244971
Page Count: 20
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Random Walks and Harmonic Functions on Infinite Planar Graphs Using Square Tilings
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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 for triangulations of a disk.

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