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.
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Supported in part by NSF Grant DMS-0504882.
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Schweinsberg, J. Loop-Erased Random Walk on Finite Graphs and the Rayleigh Process. J Theor Probab 21, 378–396 (2008). https://doi.org/10.1007/s10959-007-0125-7
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DOI: https://doi.org/10.1007/s10959-007-0125-7