  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.
