Distribution of sizes of erased loops for loop-erased random walks

Abstract

We study the distribution of sizes of erased loops for loop-erased random walks on regular and fractal lattices. We show that for arbitrary graphs the probability P(l) of generating a loop of perimeter l is expressible in terms of the probability P_st(l) of forming a loop of perimeter l when a bond is added to a random spanning tree on the same graph by the simple relation P(l)=P_st(l)/l. On d-dimensional hypercubical lattices, P(l) varies as l^-σ for large l, where σ=1+2/z for 1<d<4, where z is the fractal dimension of the loop-erased walks on the graph. On recursively constructed fractals with d-tilde < 2 this relation is modified to σ=1+2d-bar/(d-tildez), where d-bar is the Hausdorff and d-tilde is the spectral dimension of the fractal.