Loop-Erased Random Walk on Finite Graphs and the Rayleigh Process

Jason Schweinsberg ()
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Jason Schweinsberg: U.C. San Diego

Journal of Theoretical Probability, 2008, vol. 21, issue 2, 378-396

Abstract: 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.

Keywords: Loop-erased random walk; Rayleigh process; 60G50; 60K35; 60J75 (search for similar items in EconPapers)
Date: 2008
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DOI: 10.1007/s10959-007-0125-7

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