Mixing Time for Square Tilings

TL;DR

This paper studies the convergence rate of a Markov chain on square tilings of regions in the plane, proving polynomial mixing times for certain cases and exploring weighted variants with rapid mixing, supported by simulations.

Contribution

It establishes polynomial mixing times for tilings by small and large squares, including weighted cases, and introduces conjectures about frozen regions in random tilings.

Findings

Polynomial mixing time for $n imes ext{log} n$ regions with $1 imes 1$ and $s imes s$ squares.

Rapid mixing of $O(n^4 ext{log} n)$ for weighted chains under certain conditions.

Simulations suggest the existence of frozen regions in random square tilings.

Abstract

We consider tilings of $\mathbb{Z}^2$ by two types of squares. We are interested in the rate of convergence to the stationarity of a natural Markov chain defined for square tilings. The rate of convergence can be represented by the mixing time which measures the amount of time it takes the chain to be close to its stationary distribution. We prove polynomial mixing time for $n \times \log n $ regions in the case of tilings by $1 \times 1$ and $s \times s$ squares. We also consider a weighted Markov chain with weights $\lambda$ being put on big squares. We show rapid mixing of $O(n^4 \log n)$ with conditions on $\lambda$. We provide simulations that suggest different conjectures, one of which is the existence of frozen regions in random tilings by squares.