Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs

2011/07/31 by Richard W. Kenyon, David B. Wilson · 1 citation
Mathematics · #math.PR #msc:60C05 #msc:82B20 #msc:05C05 #msc:05C50

paper · pdf · doi:10.1090/s0894-0347-2014-00819-5

published as Journal of the American Mathematical Society, 28(4):985-1030, 2015 · 45 pages, many figures. v2 has an expanded introduction, a revised section on the LERW intensity, and an expanded appendix on the annular matrix

arxiv created 2014/06/22 · arxiv updated 2015/12/22

Abstract

We show how to compute the probabilities of various connection topologies for uniformly random spanning trees on graphs embedded in surfaces. As an application, we show how to compute the "intensity" of the loop-erased random walk in \mathbb Z², that is, the probability that the walk from (0,0) to infinity passes through a given vertex or edge. For example, the probability that it passes through (1,0) is 5/16; this confirms a conjecture from 1994 about the stationary sandpile density on \mathbb Z². We do the analogous computation for the triangular lattice, honeycomb lattice and \mathbb Z × \mathbb R, for which the probabilities are 5/18, 13/36, and 1/4-1/π² respectively.

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• Topology of a Uniform Spanning Tree on a Cylinder