### Abstract

We show that on every Ramanujan graph G, the simple random walk exhibits cutoff: when G has n vertices and degree d, the total-variation distance of the walk from the uniform distribution at time t=dd-2logd-1n+slogn is asymptotically P(Z>cs) where Z is a standard normal variable and c= c(d) is an explicit constant. Furthermore, for all 1 ≤ p≤ ∞, d-regular Ramanujan graphs minimize the asymptotic L^{p}-mixing time for SRW among alld-regular graphs. Our proof also shows that, for every vertex x in G as above, its distance from n- o(n) of the vertices is asymptotically log _{d-1}n.

Original language | English (US) |
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Pages (from-to) | 1190-1216 |

Number of pages | 27 |

Journal | Geometric and Functional Analysis |

Volume | 26 |

Issue number | 4 |

DOIs | |

State | Published - Jul 1 2016 |

### ASJC Scopus subject areas

- Analysis
- Geometry and Topology

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## Cite this

Lubetzky, E., & Peres, Y. (2016). Cutoff on all Ramanujan graphs.

*Geometric and Functional Analysis*,*26*(4), 1190-1216. https://doi.org/10.1007/s00039-016-0382-7