# Cutoff for Ramanujan graphs via degree inflation

## TL;DR

This paper investigates the cutoff phenomenon for random walks on Ramanujan graphs, providing a new proof under a bounded cycle condition that relates to the graph's local structure.

## Contribution

It offers a novel argument establishing cutoff for Ramanujan graphs assuming a uniform bound on the number of cycles in local neighborhoods.

## Key findings

- Cutoff occurs around the diameter lower bound in Ramanujan graphs.
- A new proof technique based on degree inflation and local cycle bounds.
- Results apply to sequences of Ramanujan graphs with controlled local complexity.

## Abstract

Recently Lubetzky and Peres showed that simple random walks on a sequence of $d$-regular Ramanujan graphs $G_n=(V_n,E_n)$ of increasing sizes exhibit cutoff in total variation around the diameter lower bound $\frac{d}{d-2}\log_{d-1}|V_n| $. We provide a different argument under the assumption that for some $r(n) \gg 1$ the maximal number of simple cycles in a ball of radius $r(n)$ in $G_n$ is uniformly bounded in $n$.