# Cutoff on Ramanujan complexes and classical groups

## TL;DR

This paper proves the total-variation cutoff phenomenon for simple random walks on Ramanujan complexes of type A_d, providing explicit generators for classical groups where cutoff occurs, advancing understanding of mixing times on expanders.

## Contribution

It establishes total-variation cutoff for random walks on Ramanujan complexes and identifies explicit generators for classical groups with this property.

## Key findings

- Proves cutoff for Ramanujan complexes of type A_d.
- Provides explicit generators for PGL_n(\u021b_q) with cutoff.
- Advances understanding of mixing times on sparse expanders.

## Abstract

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.