Cutoff on all Ramanujan graphs

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 $${t=\frac{d}{d-2} \log_{d-1} n + s\sqrt{\log n}}$$ 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 $${\log_{d-1} n}$$ .