Three lectures on random proper colorings of $\mathbb{Z}^d$

Lectures on the Spin and Loop $O(n)$ Models

The classical spin $O(n)$ model is a model on a $d$-dimensional lattice in which a vector on the $(n-1)$-dimensional sphere is assigned to every lattice site and the vectors at adjacent sites interact ferromagnetically via their inner product. Special cases include the Ising model ($n=1$), the XY model ($n=2$) and the Heisenberg model ($n=3$). We discuss questions of long-range order and decay of correlations in the spin $O(n)$ model for different combinations of the lattice dimension $d$ and the number of spin components $n$. The loop $O(n)$ model is a model for a random configuration of disj…

Proper 3-colorings of $\mathbb{Z}^2$ are Bernoulli

We consider the unique measure of maximal entropy for proper 3-colorings of $\mathbb{Z}^2$, or equivalently, the so-called zero-slope Gibbs measure. Our main result is that this measure is Bernoulli, or equivalently, that it can be expressed as the image of a translation-equivariant function of independent and identically distributed random variables placed on $\mathbb{Z}^2$. Along the way, we obtain various estimates on the mixing properties of this measure.

利用图的最大度数线性数目的颜色在线性时间内进行图的正确着色

Proper colorings of a graph in linear time using a number of colors linear in the maximum degree of the graph

本文提出了一种在最大度数线性时间内完成图着色的算法，适用于颜色数目大于3.637Δ+1的情况。

Rigidity of proper colorings of $\mathbb{Z}^d$

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 emer…

在线图上使用 $(1+o(1))Δ$ 种颜色采样正常着色

Sampling Proper Colorings on Line Graphs Using $(1+o(1))Δ$ Colors

本文证明了当颜色数 $q>(1+o(1))Δ$ 时，单点 Glauber 动力学在含 $n$ 个顶点、最大度为 $Δ$ 的线图上采样正常着色的混合时间为 $O_Δ(n\log n)$，主要利用了矩阵滴落定理完成证明。

有界度图的有限依赖随机着色

Finitely dependent random colorings of bounded degree graphs

本文证明了任意有界度图存在有限依赖的随机正常着色，并给出了颜色数上界及在幺模传递图上的自同构不变构造。