A Lower Bound for the Mixing Time of the Random-to-Random Insertions Shuffle

TL;DR

This paper establishes a new lower bound on the mixing time of the random-to-random insertions shuffle, supporting the conjecture of a cutoff at 3/4 n log n, by analyzing card positions during the shuffle.

Contribution

It provides the first lower bound matching the conjectured cutoff point, advancing understanding of the shuffle's mixing time behavior.

Findings

Lower bound of (3/4 - o(1)) n log n for mixing time

High probability of certain card position distributions

Analysis of cards yet-to-be-removed impacts measure comparison

Abstract

The best known lower and upper bounds on the mixing time for the random-to-random insertions shuffle are $(1/2-o(1))n\log n$ and $(2+o(1))n\log n$. A long standing open problem is to prove that the mixing time exhibits a cutoff. In particular, Diaconis conjectured that the cutoff occurs at $3/4n\log n$. Our main result is a lower bound of $t_n = (3/4-o(1))n\log n$, corresponding to this conjecture. Our method is based on analysis of the positions of cards yet-to-be-removed. We show that for large $n$ and $t_n$ as above, there exists $f(n)=\Theta(\sqrt{n\log n})$ such that, with high probability, under both the measure induced by the shuffle and the stationary measure, the number of cards within a certain distance from their initial position is $f(n)$ plus a lower order term. However, under the induced measure, this lower order term is strongly influenced by the number of cards yet-to-be-removed, and is of higher order than for the stationary measure.