TY - JOUR

T1 - Cutoff on all Ramanujan graphs

AU - Lubetzky, Eyal

AU - Peres, Yuval

N1 - Publisher Copyright:
© 2016, Springer International Publishing.

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.

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U2 - 10.1007/s00039-016-0382-7

DO - 10.1007/s00039-016-0382-7

M3 - Article

AN - SCOPUS:84989172099

VL - 26

SP - 1190

EP - 1216

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

SN - 1016-443X

IS - 4

ER -