Cutoff phenomena for random walks on random regular graphs

Eyal Lubetzky, Allan Sly

Research output: Contribution to journalArticle

Abstract

The cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite Markov chains to equilibrium. Many natural families of chains are believed to exhibit cutoff, and yet establishing this fact is often extremely challenging. An important such family of chains is the random walk on g(n, d), a random d-regular graph on n vertices. It is well known that almost every such graph for d ≥ 3 is an expander, and even essentially Ramanujan, implying a mixing time of O(log n). According to a conjecture of Peres, the simple random walk on g(n, d) for such d should then exhibit cutoff with high probability (whp). As a special case of this, Durrett conjectured that the mixing time of the lazy random walk on a random 3-regular graph is whp (6 + o(1)) log2 n. In this work we confirm the above conjectures and establish cutoff in total-variation, its location, and its optimal window, both for simple and for non-backtracking random walks on g(n, d). Namely, for any fixed d ≥ 3, the simple random walk on g(n, d) whp has cutoff at (d/(d - 2)) logd-1 n with window order √log n. Surprisingly, the non-backtracking random walk on g(n, d) whp has cutoff already at logd-1 n with constant window order. We further extend these results to g(n, d) for any d = no (1) that grows with n (beyond which the mixing time is O(1)), where we establish concentration of the mixing time on one of two consecutive integers.

Original languageEnglish (US)
Pages (from-to)475-510
Number of pages36
JournalDuke Mathematical Journal
Volume153
Issue number3
DOIs
StatePublished - Jun 2010

Fingerprint

Mixing Time
Regular Graph
Random Graphs
Random walk
Simple Random Walk
Expander
Ramanujan
Total Variation
Consecutive
Markov chain
Integer
Graph in graph theory
Family

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Cutoff phenomena for random walks on random regular graphs. / Lubetzky, Eyal; Sly, Allan.

In: Duke Mathematical Journal, Vol. 153, No. 3, 06.2010, p. 475-510.

Research output: Contribution to journalArticle

@article{7dd4a37e72a24b73acacb760df0d995e,
title = "Cutoff phenomena for random walks on random regular graphs",
abstract = "The cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite Markov chains to equilibrium. Many natural families of chains are believed to exhibit cutoff, and yet establishing this fact is often extremely challenging. An important such family of chains is the random walk on g(n, d), a random d-regular graph on n vertices. It is well known that almost every such graph for d ≥ 3 is an expander, and even essentially Ramanujan, implying a mixing time of O(log n). According to a conjecture of Peres, the simple random walk on g(n, d) for such d should then exhibit cutoff with high probability (whp). As a special case of this, Durrett conjectured that the mixing time of the lazy random walk on a random 3-regular graph is whp (6 + o(1)) log2 n. In this work we confirm the above conjectures and establish cutoff in total-variation, its location, and its optimal window, both for simple and for non-backtracking random walks on g(n, d). Namely, for any fixed d ≥ 3, the simple random walk on g(n, d) whp has cutoff at (d/(d - 2)) logd-1 n with window order √log n. Surprisingly, the non-backtracking random walk on g(n, d) whp has cutoff already at logd-1 n with constant window order. We further extend these results to g(n, d) for any d = no (1) that grows with n (beyond which the mixing time is O(1)), where we establish concentration of the mixing time on one of two consecutive integers.",
author = "Eyal Lubetzky and Allan Sly",
year = "2010",
month = "6",
doi = "10.1215/00127094-2010-029",
language = "English (US)",
volume = "153",
pages = "475--510",
journal = "Duke Mathematical Journal",
issn = "0012-7094",
publisher = "Duke University Press",
number = "3",

}

TY - JOUR

T1 - Cutoff phenomena for random walks on random regular graphs

AU - Lubetzky, Eyal

AU - Sly, Allan

PY - 2010/6

Y1 - 2010/6

N2 - The cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite Markov chains to equilibrium. Many natural families of chains are believed to exhibit cutoff, and yet establishing this fact is often extremely challenging. An important such family of chains is the random walk on g(n, d), a random d-regular graph on n vertices. It is well known that almost every such graph for d ≥ 3 is an expander, and even essentially Ramanujan, implying a mixing time of O(log n). According to a conjecture of Peres, the simple random walk on g(n, d) for such d should then exhibit cutoff with high probability (whp). As a special case of this, Durrett conjectured that the mixing time of the lazy random walk on a random 3-regular graph is whp (6 + o(1)) log2 n. In this work we confirm the above conjectures and establish cutoff in total-variation, its location, and its optimal window, both for simple and for non-backtracking random walks on g(n, d). Namely, for any fixed d ≥ 3, the simple random walk on g(n, d) whp has cutoff at (d/(d - 2)) logd-1 n with window order √log n. Surprisingly, the non-backtracking random walk on g(n, d) whp has cutoff already at logd-1 n with constant window order. We further extend these results to g(n, d) for any d = no (1) that grows with n (beyond which the mixing time is O(1)), where we establish concentration of the mixing time on one of two consecutive integers.

AB - The cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite Markov chains to equilibrium. Many natural families of chains are believed to exhibit cutoff, and yet establishing this fact is often extremely challenging. An important such family of chains is the random walk on g(n, d), a random d-regular graph on n vertices. It is well known that almost every such graph for d ≥ 3 is an expander, and even essentially Ramanujan, implying a mixing time of O(log n). According to a conjecture of Peres, the simple random walk on g(n, d) for such d should then exhibit cutoff with high probability (whp). As a special case of this, Durrett conjectured that the mixing time of the lazy random walk on a random 3-regular graph is whp (6 + o(1)) log2 n. In this work we confirm the above conjectures and establish cutoff in total-variation, its location, and its optimal window, both for simple and for non-backtracking random walks on g(n, d). Namely, for any fixed d ≥ 3, the simple random walk on g(n, d) whp has cutoff at (d/(d - 2)) logd-1 n with window order √log n. Surprisingly, the non-backtracking random walk on g(n, d) whp has cutoff already at logd-1 n with constant window order. We further extend these results to g(n, d) for any d = no (1) that grows with n (beyond which the mixing time is O(1)), where we establish concentration of the mixing time on one of two consecutive integers.

UR - http://www.scopus.com/inward/record.url?scp=77957796100&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957796100&partnerID=8YFLogxK

U2 - 10.1215/00127094-2010-029

DO - 10.1215/00127094-2010-029

M3 - Article

VL - 153

SP - 475

EP - 510

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 3

ER -