An O(n2) bound for the relaxation time of a Markov chain on cladograms

A cladogram is an unrooted tree with labeled leaves and unlabeled internal branchpoints of degree 3. Aldous has studied a Markov chain on the set of n-leaf cladograms in which each transition consists of removing a random leaf and its incident edge from the tree and then reattaching the leaf to a random edge of the remaining tree. Using coupling methods, Aldous showed that the relaxation time (i.e., the inverse of the spectral gap) for this chain is O!n3". Here, we use a method based on distinguished paths to prove an O!n2" bound for the relaxation time, establishing a conjecture of Aldous. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 20, 59–70, 2001