### 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 |

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### ASJC Scopus subject areas

- Analysis
- Geometry and Topology

### Cite this

*Geometric and Functional Analysis*,

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

**Cutoff on all Ramanujan graphs.** / Lubetzky, Eyal; Peres, Yuval.

Research output: Contribution to journal › Article

*Geometric and Functional Analysis*, vol. 26, no. 4, pp. 1190-1216. https://doi.org/10.1007/s00039-016-0382-7

}

TY - JOUR

T1 - Cutoff on all Ramanujan graphs

AU - Lubetzky, Eyal

AU - Peres, Yuval

PY - 2016/7/1

Y1 - 2016/7/1

N2 - 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 Lp-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-1n.

AB - 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 Lp-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-1n.

UR - http://www.scopus.com/inward/record.url?scp=84989172099&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84989172099&partnerID=8YFLogxK

U2 - 10.1007/s00039-016-0382-7

DO - 10.1007/s00039-016-0382-7

M3 - Article

VL - 26

SP - 1190

EP - 1216

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

SN - 1016-443X

IS - 4

ER -