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 $l^{-\sigma }$ for large $l$, where $\sigma =1+2/z$ for $1<d<4$, where z is the fractal dimension of the loop-erased walks on the graph. On recursively constructed fractals with $\tilde{d}<2$ this relation is modified to $\sigma =1+2\bar{d}/(\tilde{d}z)$, where $\bar{d}$ is the hausdorff and $\tilde{d}$ is the spectral dimension of the fractal.

4 pages, RevTex, 3 figures

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