方形瓷砖的混合时间
Mixing Time for Square Tilings
摘要：
我们考虑了$\mathbb{Z}^2$的两种类型的正方形的平铺。我们感兴趣的是收敛速度的平稳性的自然马尔可夫链定义的平方平铺。收敛速度可以用混合时间来表示，混合时间测量链接近其平稳分布所需的时间。我们证明了在使用$1 \times 1$和$s \times s$正方形进行镶嵌的情况下，$n \times \log n $区域的多项式混合时间。我们还考虑了一个加权马尔可夫链的权重$λ$被放在大广场。我们展示了$O(n^4 \log n)$与$λ$上的条件的快速混合。我们提供了模拟，建议不同的apturtures，其中之一是冻结区域的存在，随机平铺的正方形。
英文摘要：
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 $λ$ being put on big squares. We show rapid mixing of $O(n^4 \log n)$ with conditions on $λ$. We provide simulations that suggest different conjectures, one of which is the existence of frozen regions in random tilings by squares.
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