# Cutoff on Graphs and the Sarnak-Xue Density of Eigenvalues

## TL;DR

This paper demonstrates that the Sarnak-Xue density property, a weaker spectral condition than Ramanujan, suffices to establish cutoff phenomena in certain families of Schreier graphs, advancing understanding of eigenvalue distributions.

## Contribution

It introduces the Sarnak-Xue density condition as a new spectral criterion to prove cutoff in Schreier graphs, extending previous results beyond Ramanujan graphs.

## Key findings

- Schreier graphs of SL_2(F_t) exhibit cutoff under the Sarnak-Xue condition
- The Sarnak-Xue density property generalizes Ramanujan spectral conditions
- Supports a conjecture of Rivin and Sardari on eigenvalue distributions

## Abstract

It was recently shown by Lubetzky and Peres (2016) and by Sardari (2018) that Ramanujan graphs, i.e., graphs with the optimal spectrum, exhibit cutoff of the simple random walk in an optimal time and have an optimal almost-diameter. We show that this spectral condition can be replaced by a weaker condition, the Sarnak-Xue density property, to deduce similar results. This allows us to prove that some natural families of Schreier graphs of the $SL_2(\mathbb{F}_t)$-action on the projective line exhibit cutoff, thus proving a special case of a conjecture of Rivin and Sardari.