Mixing time of the adjacent walk on the simplex

Citation
Caputo, Pietro et al., Mixing time of the adjacent walk on the simplex, Annals of probability (Online) , 48(5), 2020, pp. 2449-2493
ISSN journal
2168894X
Volume
48
Issue
5
Year of publication
2020
Pages
2449 - 2493
Database
ACNP
SICI code
Abstract
By viewing the N-simplex as the set of positions of N.1 ordered particles on the unit interval, the adjacent walk is the continuous-time Markov chain obtained by updating independently at rate 1 the position of each particle with a sample from the uniform distribution over the interval given by the two particles adjacent to it. We determine its spectral gap and prove that both the total variation distance and the separation distance to the uniform distribution exhibit a cutoff phenomenon, with mixing times that differ by a factor 2. The results are extended to the family of log-concave distributions obtained by replacing the uniform sampling by a symmetric log-concave Beta distribution.