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 G1/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 n1/2 and (n log n)1/2; the limiting distributions are obtained as well.KeywordsMarkov ChainSpan TreeComplete GraphFactorial MomentRandom Walk AlgorithmThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.