Abstract
We give sharp rates of convergence for a natural Markov chain on the space of phylogenetic trees and dually for the natural random walk on the set of perfect matchings in the complete graph on $2n$ vertices. Roughly, the results show that $(1/2) n \log n$ steps are necessary and suffice to achieve randomness. The proof depends on the representation theory of the symmetric group and a bijection between trees and matchings.
Highlights
A phylogenetic tree with l leaves is a rooted binary tree with l labeled leaves
In hope of duplicating the achievements of comparison theory in the analysis of random walk on groups (Diaconis & Saloff-Coste (1993a), Diaconis & Saloff-Coste (1993b), Aldous & Fill (2002)) we searched for a Markov chain on trees which permits a complete analysis
We briefly describe the correspondence between matchings and trees
Summary
A phylogenetic tree with l leaves is a rooted binary tree with l labeled leaves. For example, when l = 3, there are three such distinct trees: R. A perfect matching on 2n points is a partition of 1, 2, . We analyze a natural random walk on Mn, the set of perfect matchings on 2n points, along with the isomorphic walk on trees. Theorem 1 For the Markov chain K(x, y) of (1) on Mn the space of perfect matchings on. Theorem 1 is proved in Section 3 by bounding the total variation norm by the L2 norm. This is expressed exactly in terms of symmetric group characters. To conclude this introduction, we discuss background on phylogenetic trees, random matchings, diffusion problems, zonal polynomials, and the Metropolis algorithm.
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