Mixing Time for Square Tilings

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.