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 language | English (US) |
|---|---|
| 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
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 journal › Article
}
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 -