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.
Narutaka Ozawa. "An entropic proof of cutoff on Ramanujan graphs." Electron. Commun. Probab. 25 1 - 8, 2020. https://doi.org/10.1214/20-ECP358