### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 419-435 |

Number of pages | 17 |

Journal | Electronic Journal of Probability |

Volume | 16 |

DOIs | |

State | Published - Jan 1 2011 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- Cutoff phenomenon
- Expander graphs
- Explicit constructions
- Random walks

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## Cite this

*Electronic Journal of Probability*,

*16*, 419-435. https://doi.org/10.1214/EJP.v16-869