Convergence of the Length of the Loop-erased Random Walk on Finite Graphs to the Rayleigh Process
نویسنده
چکیده
Let (Gn) ∞ n=1 be a sequence of finite graphs, and let Yt be the length of a loop-erased random walk on Gn after t steps. We show that for a large family of sequences of finite graphs, which includes the case in which Gn is the d-dimensional torus of size-length n for d ≥ 4, the process (Yt) ∞ 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 looperased random walks to show that the uniform spanning tree on large finite graphs converges to the Brownian continuum random tree of Aldous.
منابع مشابه
Loop-erased Random Walk on Finite Graphs and the Rayleigh Process
Let (Gn) ∞ n=1 be a sequence of finite graphs, and let Yt be the length of a loop-erased random walk on Gn after t steps. We show that for a large family of sequences of finite graphs, which includes the case in which Gn is the d-dimensional torus of size-length n for d ≥ 4, the process (Yt) ∞ t=0, suitably normalized, converges to the Rayleigh process introduced by Evans, Pitman, and Winter. O...
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