Note on Exact and Asymptotic Distributions of the Parameters of the Loop-Erased Random Walk on the Complete Graph
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 Chain Span Tree Complete Graph Factorial Moment Random Walk Algorithm
Unable to display preview. Download preview PDF.
- A. Broder Generating random spanning trees Foundations of Computer Sciencer (1989), 442–447.Google Scholar
- D. B. Wilson Generating random spanning trees more quickly than the cover time Proceedings of the Twenty-Eigth Annual ACM Symposium on the Theory of Computing (1996), ACM, New York, 296–303.Google Scholar