Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Comparing mixing times on sparse random graphs

Anna Ben-Hamou, Eyal Lubetzky, and Yuval Peres

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Abstract

It is natural to expect that nonbacktracking random walk will mix faster than simple random walks, but so far this has only been proved in regular graphs. To analyze typical irregular graphs, let $G$ be a random graph on $n$ vertices with minimum degree 3 and a degree distribution that has exponential tails. We determine the precise worst-case mixing time for simple random walk on $G$, and show that, with high probability, it exhibits cutoff at time ${\mathbf{h}}^{-1}\log n$, where ${\mathbf{h}}$ is the asymptotic entropy for simple random walk on a Galton–Watson tree that approximates $G$ locally. (Previously this was only known for typical starting points.) Furthermore, we show this asymptotic mixing time is strictly larger than the mixing time of nonbacktracking walk, via a delicate comparison of entropies on the Galton–Watson tree.

Résumé

Il est naturel de s’attendre à ce que la marche aléatoire sans rebroussement mélange plus vite que la marche aléatoire simple, mais jusqu’ici, cela n’était prouvé que dans le cas des graphes réguliers. Pour analyser le cas de graphes irréguliers typiques, soit $G$ un graphe aléatoire à $n$ sommets de degrés au moins $3$ et distribués selon une loi à queue exponentielle. On détermine le temps de mélange partant du pire point de départ pour la marche aléatoire simple sur $G$, et l’on montre qu’avec grande probabilité, cette marche présente le phénomène de cutoff au temps ${\mathbf{h}}^{-1}\log n$, où ${\mathbf{h}}$ est l’entropie asymptotique de la marche aléatoire simple sur un arbre de Galton–Watson qui est une approximation locale de $G$. (Précédemment, cela n’était connu que pour des points de départ typiques.) De plus, on montre que ce temps de mélange est strictement plus grand que celui de la marche aléatoire sans rebroussement, via une comparison délicate des entropies sur l’arbre de Galton–Watson.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 1116-1130.

Dates
Received: 28 July 2017
Revised: 3 April 2018
Accepted: 24 April 2018
First available in Project Euclid: 14 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1557820844

Digital Object Identifier
doi:10.1214/18-AIHP911

Zentralblatt MATH identifier
1403.05133

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 05C80: Random graphs [See also 60B20]

Keywords
Random graphs Random walks Nonbacktracking vs. Simple random walk Mixing times of Markov chains

Citation

Ben-Hamou, Anna; Lubetzky, Eyal; Peres, Yuval. Comparing mixing times on sparse random graphs. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 1116--1130. doi:10.1214/18-AIHP911. https://projecteuclid.org/euclid.aihp/1557820844


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