Mathematics and Computer Science II pp 423-440 | Cite as

# Note on Exact and Asymptotic Distributions of the Parameters of the Loop-Erased Random Walk on the Complete Graph

## Abstract

We study the loop-erased random walk algorithm for generating a random spanning tree of the complete graph on n vertices. The number of moves is shown to be distributed as n − 2 plus G_{1/n}, a Geometric with expectation n. The lengths of the paths (branches) that are added to a subtree are jointly distributed as the consecutive waiting times for heads in a sequence of time-biased, but independent, coin flips. As a corollary, the subtree size is shown to grow, with high probability, at the rate (rn)^{1/2}, r being the number of branches added. The lengths of the largest path and the largest loop are shown to scale with n^{1/2} and (n log n)^{1/2}; the limiting distributions are obtained as well.

## Keywords

Markov Chain Span Tree Complete Graph Factorial Moment Random Walk Algorithm## Preview

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