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

TL;DR

This paper investigates the distribution of erased loop sizes in loop-erased random walks on various graphs, revealing power-law behaviors and relations to graph dimensions, with implications for understanding complex network structures.

Contribution

It derives a general relation between loop size probabilities and spanning trees, and extends the analysis to fractal lattices with unique dimensional properties.

Findings

Probability of loop size follows a power-law distribution with exponent depending on graph dimensions.

Derived a universal relation connecting loop size distribution to spanning tree probabilities.

Identified modifications of the power-law exponent for fractal lattices with spectral dimension less than 2.

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.