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. 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.