Cutoff on All Ramanujan Graphs
نویسندگان
چکیده
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 = d d−2 logd−1 n + s √ logn is asymptotically P(Z > c s) 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-mixing time for SRW among all d-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 logd−1 n.
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