The Annals of Probability

Cutoff for nonbacktracking random walks on sparse random graphs

Anna Ben-Hamou and Justin Salez

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Abstract

A finite ergodic Markov chain exhibits cutoff if its distance to stationarity remains close to 1 over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Discovered in the context of card shuffling (Aldous–Diaconis, 1986), this phenomenon is now believed to be rather typical among fast mixing Markov chains. Yet, establishing it rigorously often requires a challengingly detailed understanding of the underlying chain. Here, we consider nonbacktracking random walks on random graphs with a given degree sequence. Under a general sparsity condition, we establish the cutoff phenomenon, determine its precise window and prove that the cutoff profile approaches a remarkably simple, universal shape.

Article information

Source
Ann. Probab., Volume 45, Number 3 (2017), 1752-1770.

Dates
Received: April 2015
Revised: January 2016
First available in Project Euclid: 15 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1494835230

Digital Object Identifier
doi:10.1214/16-AOP1100

Mathematical Reviews number (MathSciNet)
MR3650414

Zentralblatt MATH identifier
1372.60101

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60G50: Sums of independent random variables; random walks 05C80: Random graphs [See also 60B20] 05C81: Random walks on graphs

Keywords
Cutoff phenomenon nonbacktracking random walks random graphs

Citation

Ben-Hamou, Anna; Salez, Justin. Cutoff for nonbacktracking random walks on sparse random graphs. Ann. Probab. 45 (2017), no. 3, 1752--1770. doi:10.1214/16-AOP1100. https://projecteuclid.org/euclid.aop/1494835230


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