An entropic proof of cutoff on Ramanujan graphs

TL;DR

This paper presents a new proof of the cutoff phenomenon for simple random walks on Ramanujan graphs, using functional analysis and entropy, offering an alternative to existing proofs.

Contribution

It introduces an entropic proof method for the cutoff phenomenon on Ramanujan graphs, expanding the analytical tools available for such proofs.

Findings

Proof of cutoff phenomenon using entropic methods

Alternative approach based on functional analysis

Reinforces understanding of random walk mixing on Ramanujan graphs

Abstract

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.