$\mathbb{Z}^d$真着色的刚性
Rigidity of proper colorings of $\mathbb{Z}^d$
摘要：
在$\mathbb{Z}^d$中，域的适当$q$着色是将$q$颜色之一分配给域的每个顶点以使得邻顶点被不同地着色的函数。对一个适当的$q$ -随机均匀着色进行抽样，着色是否通常表现出长程有序？自从多布鲁申的工作以来，人们已经知道，当$q$比$d$大时，就不会出现这样的排序。我们证明了当$d$足够高时，每个$q$都存在长程有序，并进一步刻画了该模型的所有周期最大熵吉布斯态.有序也出现在低维，如果晶格$\mathbb{Z}^d$被替换为$\mathbb{Z}^{d_1}\times\mathbb{T}^{d_2}$与$d_1\ge 2$，$d=d_1+d_2$足够高和$\mathbb{T}$周期的均匀长度。这些结果回答了Berker-Kadanoff（1980）、Kotecklord（1985）和Salas-Sokal（1997）提出的问题。
英文摘要：
A proper $q$-coloring of a domain in $\mathbb{Z}^d$ is a function assigning one of $q$ colors to each vertex of the domain such that adjacent vertices are colored differently. Sampling a proper $q$-coloring uniformly at random, does the coloring typically exhibit long-range order? It has been known since the work of Dobrushin that no such ordering can arise when $q$ is large compared with $d$. We prove here that long-range order does arise for each $q$ when $d$ is sufficiently high, and further characterize all periodic maximal-entropy Gibbs states for the model. Ordering is also shown to emerge in low dimensions if the lattice $\mathbb{Z}^d$ is replaced by $\mathbb{Z}^{d_1}\times\mathbb{T}^{d_2}$ with $d_1\ge 2$, $d=d_1+d_2$ sufficiently high and $\mathbb{T}$ a cycle of even length. The results address questions going back to Berker--Kadanoff (1980), Kotecký (1985) and Salas--Sokal (1997).
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