Cutoff for Ramanujan graphs via degree inflation

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 distan…

Cutoff on Ramanujan complexes and classical groups

The total-variation cutoff phenomenon has been conjectured to hold for simple random walk on all transitive expanders. However, very little is actually known regarding this conjecture, and cutoff on sparse graphs in general. In this paper we establish total-variation cutoff for simple random walk on Ramanujan complexes of type $\widetilde{A}_{d}$ $(d\geq1)$. As a result, we obtain explicit generators for the finite classical groups $\mathrm{PGL}_{n}(\mathbb{F}_{q})$ for which the associated Cayley graphs exhibit total-variation cutoff.

Bounded cutoff window for the non-backtracking random walk on Ramanujan Graphs

We prove that the non-backtracking random walk on Ramanujan graphs with large girth exhibits the fastest possible cutoff with a bounded window.

An entropic proof of cutoff on Ramanujan graphs

It is recently proved by Lubetzky and Peres that the simple random walk on a Ramanujan graph exhibits a cutoff phenomenon, that is to say, the total variation distance of the random walk distribution from the uniform distribution drops abruptly from near $1$ to near $0$. There are already a few alternative proofs of this fact. In this note, we give yet another proof based on functional analysis and entropic consideration.

Random walks on Ramanujan complexes and digraphs

The cutoff phenomenon was recently confirmed for random walks on Ramanujan graphs by the first author and Peres. In this work, we obtain analogs in higher dimensions, for random walk operators on any Ramanujan complex associated with a simple group $G$ over a local field $F$. We show that if $T$ is any $k$-regular $G$-equivariant operator on the Bruhat-Tits building with a simple combinatorial property (collision-free), the associated random walk on the $n$-vertex Ramanujan complex has cutoff at time $\log_k n$. The high dimensional case, unlike that of graphs, requires tools from non-commutat…

长程扩张图：构造与截断

Long-range expanders: construction and cutoff

本文证明每个Ramanujan图具有长程扩张，构造对数围长的显式长程扩张图，并确立固定度长程扩张图序列上随机游走的截断现象及其高斯极限轮廓，为传递谱扩张图的截断猜想提供中间进展。