Duke Mathematical Journal

Cutoff phenomena for random walks on random regular graphs

Eyal Lubetzky and Allan Sly

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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 d3 is an expander, and even essentially Ramanujan, implying a mixing time of O(logn). 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))log2n. 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 d3, the simple random walk on G(n,d) whp has cutoff at (d/(d2))logd1n with window order logn. Surprisingly, the non-backtracking random walk on G(n,d) whp has cutoff already at logd1n 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

Article information

Source
Duke Math. J., Volume 153, Number 3 (2010), 475-510.

Dates
First available in Project Euclid: 4 June 2010

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1275671395

Digital Object Identifier
doi:10.1215/00127094-2010-029

Mathematical Reviews number (MathSciNet)
MR2667423

Zentralblatt MATH identifier
1202.60012

Subjects
Primary: 60B10: Convergence of probability measures 60G50: Sums of independent random variables; random walks
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 05C80: Random graphs [See also 60B20]

Citation

Lubetzky, Eyal; Sly, Allan. Cutoff phenomena for random walks on random regular graphs. Duke Math. J. 153 (2010), no. 3, 475--510. doi:10.1215/00127094-2010-029. https://projecteuclid.org/euclid.dmj/1275671395


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