Loop-erased random walk on finite graphs and the Rayleigh process

Details Loop-erased random walk on finite graphs and the Rayleigh process Loop-erased random walk on finite graphs and the Rayleigh process

2006-11-06

arxiv.org/abs/math/0611155v2

Mathematics

Probability

Scientific paper

Let $(G_n)_{n=1}^{\infty}$ 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 $d \geq 4$, the process $(Y_t)_{t=0}^{\infty}$, 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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