Electronic Journal of Probability

On sensitivity of mixing times and cutoff

Jonathan Hermon and Yuval Peres

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Abstract

A sequence of chains exhibits (total variation) cutoff (resp., pre-cutoff) if for all $0<\epsilon < 1/2$, the ratio $t_{\mathrm{mix} }^{(n)}(\epsilon )/t_{\mathrm{mix} }^{(n)}(1-\epsilon )$ tends to 1 as $n \to \infty $ (resp., the $\limsup $ of this ratio is bounded uniformly in $\epsilon $), where $t_{\mathrm{mix} }^{(n)}(\epsilon )$ is the $\epsilon $-total variation mixing time of the $n$th chain in the sequence. We construct a sequence of bounded degree graphs $G_n$, such that the lazy simple random walks (LSRW) on $G_n$ satisfy the “product condition” $\mathrm{gap} (G_n) t_{\mathrm{mix} }^{(n)}(\epsilon ) \to \infty $ as $n \to \infty $, where $\mathrm{gap} (G_n)$ is the spectral gap of the LSRW on $G_n$ (a known necessary condition for pre-cutoff that is often sufficient for cutoff), yet this sequence does not exhibit pre-cutoff.

Recently, Chen and Saloff-Coste showed that total variation cutoff is equivalent for the sequences of continuous-time and lazy versions of some given sequence of chains. Surprisingly, we show that this is false when considering separation cutoff.

We also construct a sequence of bounded degree graphs $G_n=(V_{n},E_{n})$ that does not exhibit cutoff, for which a certain bounded perturbation of the edge weights leads to cutoff and increases the order of the mixing time by an optimal factor of $\Theta (\log |V_n|)$. Similarly, we also show that “lumping” states together may increase the order of the mixing time by an optimal factor of $\Theta (\log |V_n|)$. This gives a negative answer to a question asked by Aldous and Fill.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 25, 34 pp.

Dates
Received: 30 November 2016
Accepted: 27 February 2018
First available in Project Euclid: 15 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1521079338

Digital Object Identifier
doi:10.1214/18-EJP154

Mathematical Reviews number (MathSciNet)
MR3779818

Zentralblatt MATH identifier
1387.60112

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
reversible Markov chains simple random walk mixing time cutoff, pre-cutoff perturbations sensitivity of cutoff separation cutoff counter-example

Rights
Creative Commons Attribution 4.0 International License.

Citation

Hermon, Jonathan; Peres, Yuval. On sensitivity of mixing times and cutoff. Electron. J. Probab. 23 (2018), paper no. 25, 34 pp. doi:10.1214/18-EJP154. https://projecteuclid.org/euclid.ejp/1521079338


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