# Cutoff for random to random card shuffle

## TL;DR

This paper establishes the precise cutoff time for the random to random card shuffle using eigenvalue analysis, confirming a conjecture and providing sharp bounds on the mixing time.

## Contribution

It proves the cutoff phenomenon for the random to random shuffle at a specific time, combining eigenvalue bounds with existing lower bounds to confirm the conjecture.

## Key findings

- Cutoff occurs at (3/4) n log n - (1/4) n log log n.
- Mixing time has a window of order n.
- Confirms a conjecture of Diaconis.

## Abstract

In this paper, we use the eigenvalues of the random to random card shuffle to prove a sharp upper bound for the total variation mixing time. Combined with the lower bound due to Subag, we prove that this walk exhibits cutoff at $\frac{3}{4} n \log n - \frac{1}{4}n\log\log{n}$ with window of order $n$, answering a conjecture of Diaconis.