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

  • Boris Pittel
Conference paper
Part of the Trends in Mathematics book series (TM)


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.


Markov Chain Span Tree Complete Graph Factorial Moment Random Walk Algorithm 
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Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • Boris Pittel
    • 1
    • 2
  1. 1.Ohio State UniversityColumbusUSA
  2. 2.Department of MathematicsThe Ohio State UniversityColumbusUSA

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