MIXING TIME OF THE ADJACENT WALK ON THE SIMPLEX

• 成果类型: Article
• 署名作者: Caputo, Pietro; Labbe, Cyril; Lacoin, Hubert
• 署名单位: Roma Tre University; Universite PSL; Universite Paris-Dauphine; Centre National de la Recherche Scientifique (CNRS); Instituto Nacional de Matematica Pura e Aplicada (IMPA)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/20-AOP1428
• 发表日期: 2020
• 页码: 2449-2493
• 关键词: cutoff phenomenon; spectral gap

摘要

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.