Kagome格上的混合时间
Mixing Time on the Kagome Lattice
摘要：
我们考虑一个封闭区域的Kagome格（分区的平面成正六边形和等边三角形，使每个边缘是由一个三角形和一个六边形共享）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上定义的一组Kagome平铺。收敛速度可以用混合时间来表示，混合时间衡量链接近其平稳分布所需的时间。我们得到了一个$\mathcal{O}(N^4)$上界的加权版本的自然马尔可夫链的混合时间。我们还考虑Kagome平铺限制到两个prototiles，证明翻转连接，并绘制一个$\mathcal{O}(N^4)$上界，以及在一般情况下（非加权）的自然马尔可夫链的混合时间。最后，我们提出了模拟，表明存在一个长期的现象。
英文摘要：
We consider tilings of a closed region of the Kagome lattice (partition of the plane into regular hexagons and equilateral triangles such that each edge is shared by one triangle and one hexagon). We are interested in the rate of convergence to the stationarity of a natural Markov chain defined on the set of Kagome tilings. The rate of convergence can be represented by the mixing time which mesures the amount of time it takes the chain to be close to its stationary distribution. We obtain a $\mathcal{O}(N^4)$ upper bound on the mixing time of a weighted version of the natural Markov chain. We also consider Kagome tilings restrained to two prototiles, prove flip-connectivity and draw a $\mathcal{O}(N^4)$ upper bound as well on the mixing time of the natural Markov chain in a general (non weighted) case. Finally, we present simulations that suggest existence of a long range phenomenon.
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