  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.
