Cutoff on all Ramanujan graphs

Eyal Lubetzky, Yuval Peres

Research output: Contribution to journalArticle

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 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.

Original languageEnglish (US)
Pages (from-to)1190-1216
Number of pages27
JournalGeometric and Functional Analysis
Volume26
Issue number4
DOIs
StatePublished - Jul 1 2016

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Ramanujan Graphs
Regular Graph
Total Variation Distance
Mixing Time
Simple Random Walk
Uniform distribution
Walk
Minimise
Vertex of a graph

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology

Cite this

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

In: Geometric and Functional Analysis, Vol. 26, No. 4, 01.07.2016, p. 1190-1216.

Research output: Contribution to journalArticle

Lubetzky, Eyal ; Peres, Yuval. / Cutoff on all Ramanujan graphs. In: Geometric and Functional Analysis. 2016 ; Vol. 26, No. 4. pp. 1190-1216.
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