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

By Richard Kenyon, David Bruce Wilson. 2011

47 citations recorded by the source corpus.

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 ${\mathbb Z}^2$, 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 ${\mathbb Z}^2$. We do the analogous computation for the triangular lattice, honeycomb lattice and ${\mathbb Z} \times {\mathbb R}$, for which the probabilities are 5/18, 13/36, and $1/4-1/\pi^2$ respectively.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f7735b141cf6eda25463a66bcdf4eb4649d4592c

Citations in the corpus

• Probability on Trees and Networks (2017, unknown): “The Kesten-Stigum Theorem (1966). The following are equivalent when 1 < m < ∞: (i) P[W = 0] = q ; (ii) E[W ] = 1 ; (iii) E[L log+ L] < ∞ .”