Loop-Erased Random Walk on Finite Graphs and the Rayleigh Process |
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Authors: | Jason Schweinsberg |
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Institution: | (1) Department of Mathematics, U.C. San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0112, USA |
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Abstract: | Let (G
n
)
n=1∞ be a sequence of finite graphs, and let Y
t
be the length of a loop-erased random walk on G
n
after t steps. We show that for a large family of sequences of finite graphs, which includes the case in which G
n
is the d-dimensional torus of size-length n for d≥4, the process (Y
t
)
t=0∞, suitably normalized, converges to the Rayleigh process introduced by Evans, Pitman, and Winter. Our proof relies heavily
on ideas of Peres and Revelle, who used loop-erased random walks to show that the uniform spanning tree on large finite graphs
converges to the Brownian continuum random tree of Aldous.
Supported in part by NSF Grant DMS-0504882. |
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Keywords: | Loop-erased random walk Rayleigh process |
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