Mixing time of the torus shuffle

TL;DR

This paper introduces a new theorem linking the mixing time of a card shuffle to triplet-based conditions and applies it to analyze a classic grid shuffle model, nearly confirming a long-standing conjecture.

Contribution

The paper presents a novel reduction technique for bounding mixing times using triplet conditions and applies it to a well-known card shuffling model, advancing understanding of its mixing behavior.

Findings

Bound on mixing time within polylog factor of conjecture

Reduction theorem simplifies analysis of card shuffles

Application to Diaconis's grid shuffle model

Abstract

We prove a theorem that reduces bounding the mixing time of a card shuffle to verifying a condition that involves only triplets of cards. Then we use it to analyze a classic model of card shuffling. In 1988, Diaconis introduced the following Markov chain. Cards are arranged in an $n$ by $n$ grid. Each step, choose a row or column, uniformly at random, and cyclically rotate it by one unit in a random direction. He conjectured that the mixing time is ${\rm O}(n^3 \log n)$. We obtain a bound that is within a poly log factor of the conjecture.