Cutoff on all Ramanujan graphs

TL;DR

This paper proves that simple random walks on Ramanujan graphs exhibit cutoff behavior with precise asymptotics, and that these graphs minimize mixing times among all regular graphs, revealing deep spectral and geometric properties.

Contribution

It establishes cutoff for SRW on Ramanujan graphs with explicit asymptotic formulas and shows their optimality in minimizing $L^p$-mixing times among regular graphs.

Findings

Cutoff occurs at a specific logarithmic time with normal fluctuations.

Ramanujan graphs minimize asymptotic $L^p$-mixing times among $d$-regular graphs.

Vertices are asymptotically uniformly distributed in terms of distance from a fixed vertex.

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 $\log_{d-1} n$.