### 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)) log_{2} 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)) log_{d-1} n with window order √log n. Surprisingly, the non-backtracking random walk on g(n, d) whp has cutoff already at log_{d-1} n with constant window order. We further extend these results to g(n, d) for any d = n^{o} ^{(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 language | English (US) |
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Pages (from-to) | 475-510 |

Number of pages | 36 |

Journal | Duke Mathematical Journal |

Volume | 153 |

Issue number | 3 |

DOIs | |

State | Published - Jun 2010 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Duke Mathematical Journal*,

*153*(3), 475-510. https://doi.org/10.1215/00127094-2010-029

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

Research output: Contribution to journal › Article

*Duke Mathematical Journal*, vol. 153, no. 3, pp. 475-510. https://doi.org/10.1215/00127094-2010-029

}

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

AN - SCOPUS:77957796100

VL - 153

SP - 475

EP - 510

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 3

ER -