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Cutoff on all Ramanujan graphs


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 \({t=\frac{d}{d-2} \log_{d-1} n + s\sqrt{\log n}}\) is asymptotically \({{\mathbb{P}}(Z > c \, s)}\) where \({Z}\) is a standard normal variable and \({c=c(d)}\) is an explicit constant. Furthermore, for all \({1 \leq p \leq \infty}\), \({d}\)-regular Ramanujan graphs minimize the asymptotic \({L^p}\)-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 \({\log_{d-1} n}\).

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Correspondence to Yuval Peres.

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Lubetzky, E., Peres, Y. Cutoff on all Ramanujan graphs. Geom. Funct. Anal. 26, 1190–1216 (2016).

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