The mixing time of the fifteen puzzle

TL;DR

This paper establishes bounds on the mixing time of the fifteen puzzle on an n by n torus, demonstrating it grows roughly as n^4 log n to n^4 log^2 n, revealing how quickly the puzzle randomizes.

Contribution

It provides the first rigorous bounds on the mixing time of the fifteen puzzle on a torus, quantifying its convergence rate.

Findings

Lower bound: mixing time is at least proportional to n^4 log n

Upper bound: mixing time is at most proportional to n^4 log^2 n

Constants c and C are universal and positive

Abstract

We show that there are universal positive constants c and C such that the mixing time T_{mix} for the fifteen puzzle in an n by n torus satisfies cn^4 log n < T_{mix} < Cn^4 log^2 n.