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.
Citation
Persi Diaconis. Susan Holmes. "Random Walks on Trees and Matchings." Electron. J. Probab. 7 1 - 17, 2002. https://doi.org/10.1214/EJP.v7-105
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