Open Access
September 2020 Mixing time of the adjacent walk on the simplex
Pietro Caputo, Cyril Labbé, Hubert Lacoin
Ann. Probab. 48(5): 2449-2493 (September 2020). DOI: 10.1214/20-AOP1428


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.


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Pietro Caputo. Cyril Labbé. Hubert Lacoin. "Mixing time of the adjacent walk on the simplex." Ann. Probab. 48 (5) 2449 - 2493, September 2020.


Received: 1 April 2019; Revised: 1 January 2020; Published: September 2020
First available in Project Euclid: 23 September 2020

MathSciNet: MR4152648
Digital Object Identifier: 10.1214/20-AOP1428

Primary: 60J25
Secondary: 37A25 , 82C22

Keywords: adjacent walk , Cutoff , mixing time , spectral gap

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 5 • September 2020
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