The \(L^{2}\)-cutoff for reversible Markov processes
type: Item
instance of: scholarly article
title: The \(L^{2}\)-cutoff for reversible Markov processes
zbMATH Open document ID: 1192.60088
author: Guan-Yu Chen
author: Laurent Saloff-Coste
published in: Journal of Functional Analysis
publication date: 2010-03-17T00:00:00Z
review text: The concept of cutoff was introduced by D. Aldous and P. Diaconis in a series of papers to capture the fact that, for many ergodic Markov processes, the convergence to stationary behavior occurs over a period of time that is very short compared to the waiting time for the onset of such behavior. The aim of the present paper is twofold. The first one is to establish a criterion for the existence of an \(L^2\)--cutoff for families of Markov processes starting from specific initial distributions when the associated semigroup is normal. The second one is to derive formulas for the \(L^2\)--cutoff time sequence using spectral information. A typical result reads as follows: Let \(\Omega = \{ 0, 1, \ldots \}\) and \(K\) be the Markov kernel of the birth and death chain on \(\Omega \) with uniform birth rate \(p \in (0, 1/2)\), uniform death rate \(1 - p\) and \(K(0, 0) = 1 - p\). Let \((x_n)\) be a sequence of states in \(\Omega \). Then, the discrete time family of birth and death chains with respective starting states \(x_1, x_2, \ldots \) present an \(L^2\)--cutoff if and only if \(x_n\) tends to infinity. Moreover, if there is a cutoff then \[ t_n = \frac{\log (1 - p) - \log p}{- \log (4 p (1 - p))} x_n \] is a cutoff time sequence as \(n \rightarrow \infty \). The results are illustrated by several examples, including the Ehrenfest process and the biased \((p, q)\)--random walk on the non-negative integers.
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 60J25
zbMATH DE Number: 5682617
zbMATH Keywords: \(L^2\)--cutoff
zbMATH Keywords: Markov semigroups
zbMATH Keywords: normal operators
reviewed by: Liliana Popa
MaRDI profile type: Publication
full work available at URL: j.jfa.2009.10.017
OpenAlex ID: W2081988389
cites work: Q3660628
cites work: Shuffling Cards and Stopping Times
cites work: Strong uniform times and finite random walks
cites work: The cutoff phenomenon for ergodic Markov processes
cites work: The cutoff phenomenon in finite Markov chains.
cites work: Q3138330
cites work: Separation cut-offs for birth and death chains
cites work: Q5538132
cites work: Q3484138
cites work: Random walks
cites work: Ehrenfest urn models
cites work: Q3289163
cites work: Q3549475
cites work: Q4146667
cites work: Q4371374
cites work: Q4340096
cites work: Q4358811
cites work: Q4488963
cites work: Q5846853
DOI: 10.1016/J.JFA.2009.10.017
description: scientific article