Mixing time of the adjacent walk on the simplex

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.

References in corpus (7)

• Mixing times of lozenge tiling and card shuffling Markov chains
• Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion
• On the spectral gap of the Kac walk and other binary collision processes
• A Gibbs sampler on the $n$-simplex
• Cut-off phenomenon in the uniform plane Kac walk
• Total Variation and Separation Cutoffs are not equivalent and neither one implies the other
• Mixing time and cutoff for the weakly asymmetric simple exclusion process

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• Spectral gap of the symmetric inclusion process
• Estimating MCMC convergence rates using common random number simulation