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

Invariant colorings of random planar maps

Consider Bernoulli(1/2) percolation on $\mathbb{Z}^d$, and define a perfect matching between open and closed vertices in a way that is a deterministic equivariant function of the configuration. We want to find such matching rules that make the probability that the pair of the origin is at distance greater than $r$ decay as fast as possible. For two dimensions, we give a matching of decay $cr^{1/2}$, which is optimal. For dimension at least 3 we give a matching rule that has an exponential tail. This substantially improves previous bounds. The construction has two major parts: first we define a…

Recent progress on Bernoulli convolutions

The Bernoulli convolution with parameter $λ\in(0,1)$ is the measure on $\bf R$ that is the distribution of the random power series $\sum\pmλ^n$, where $\pm$ are independent fair coin-tosses. This paper surveys recent progress on our understanding of the regularity properties of these measures.

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…

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。

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。

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

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 …