论文标题
$ \ mathbb {z}^d $的随机颜色的三个讲座
Three lectures on random proper colorings of $\mathbb{Z}^d$
论文作者
论文摘要
图形的适当$ q $颜色是对图的每个顶点的$ Q $颜色之一的分配，以使相邻的顶点对颜色有所不同。在整数晶格$ \ mathbb {z}^d $中，在所有适当的$ q $颜色中均匀地示例。获得的随机着色是否表现出任何大规模结构？它有快速的相关性衰减吗？我们讨论这些问题以及它们的答案取决于尺寸$ d $和颜色$ Q $的数量。这些问题是由统计物理学（抗铁磁材料，方冰），组合物（正确的着色，独立集）和随机Lipschitz在晶格上功能的研究来激励的。讨论介绍了一套多种工具，可用于此目的以及其他问题，包括空间混合，熵和耦合方法，吉布斯测量及其分类和精制轮廓分析。
A proper $q$-coloring of a graph is an assignment of one of $q$ colors to each vertex of the graph so that adjacent vertices are colored differently. Sample uniformly among all proper $q$-colorings of a large discrete cube in the integer lattice $\mathbb{Z}^d$. Does the random coloring obtained exhibit any large-scale structure? Does it have fast decay of correlations? We discuss these questions and the way their answers depend on the dimension $d$ and the number of colors $q$. The questions are motivated by statistical physics (anti-ferromagnetic materials, square ice), combinatorics (proper colorings, independent sets) and the study of random Lipschitz functions on a lattice. The discussion introduces a diverse set of tools, useful for this purpose and for other problems, including spatial mixing, entropy and coupling methods, Gibbs measures and their classification and refined contour analysis.
