Electronic Journal of Probability

Explicit Expanders with Cutoff Phenomena

Eyal Lubetzky and Allan Sly

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The cutoff phenomenon describes a sharp transition in the convergence of an ergodic finite Markov chain to equilibrium. Of particular interest is understanding this convergence for the simple random walk on a bounded-degree expander graph. The first example of a family of bounded-degree graphs where the random walk exhibits cutoff in total-variation was provided only very recently, when the authors showed this for a typical random regular graph. However, no example was known for an explicit (deterministic) family of expanders with this phenomenon. Here we construct a family of cubic expanders where the random walk from a worst case initial position exhibits total-variation cutoff. Variants of this construction give cubic expanders without cutoff, as well as cubic graphs with cutoff at any prescribed time-point.

Article information

Electron. J. Probab., Volume 16 (2011), paper no. 15, 419-435.

Accepted: 24 February 2011
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B10: Convergence of probability measures
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60G50: Sums of independent random variables; random walks 05C81: Random walks on graphs

Cutoff phenomenon Random walks Expander graphs Explicit constructions

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Lubetzky, Eyal; Sly, Allan. Explicit Expanders with Cutoff Phenomena. Electron. J. Probab. 16 (2011), paper no. 15, 419--435. doi:10.1214/EJP.v16-869. https://projecteuclid.org/euclid.ejp/1464820184

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