We prove that proper coloring distinguishes between block factors and finitely dependent stationary processes. A stochastic process is finitely dependent if variables at sufficiently well-separated locations are independent; it is a block factor if it can be expressed as an equivariant finite-range function of independent variables. The problem of finding non-block-factor finitely dependent processes dates back to 1965. The first published example appeared in 1993, and we provide arguably the first natural examples. Schramm proved in 2008 that no stationary 1-dependent 3-coloring of the integers exists, and asked whether a $k$-dependent $q$-coloring exists for any $k$ and $q$. We give a complete answer by constructing a 1-dependent 4-coloring and a 2-dependent 3-coloring. Our construction is canonical and natural, yet very different from all previous schemes. In its pure form it yields precisely the two finitely dependent colorings mentioned above, and no others. The processes provide unexpected connections between extremal cases of the Lovász local lemma and descent and peak sets of random permutations. Neither coloring can be expressed as a block factor, nor as a function of a finite-state Markov chain; indeed, no stationary finitely dependent coloring can be so expressed. We deduce extensions involving $d$ dimensions and shifts of finite type; in fact, any nondegenerate shift of finite type also distinguishes between block factors and finitely dependent processes.

中文翻译：

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完全依赖的着色

我们证明了适当的着色可以区分块因子和有限依赖的平稳过程。如果在充分分离的位置上的变量是独立的，则随机过程是有限依赖的；如果它可以表示为自变量的等变有限范围函数，则它是一个块因子。寻找非块因子有限依赖过程的问题可以追溯到 1965 年。第一个发布的示例出现在 1993 年，我们提供了可以说是第一个自然示例。Schramm 在 2008 年证明不存在整数的平稳 1 依赖 3 着色，并询问是否存在$k$依赖的$q$-着色存在于任何$k$和$q$. 我们通过构造一个依赖 1 的 4 着色和一个依赖 2 的 3 着色来给出一个完整的答案。我们的建筑是规范和自然的，但与以前的所有方案都有很大不同。在其纯粹的形式中，它恰好产生上述两种有限依赖的颜色，而不是其他颜色。这些过程提供了 Lovász 局部引理的极端情况与随机排列的下降和峰值集之间的意外联系。着色既不能表示为块因子，也不能表示为有限状态马尔可夫链的函数；实际上，没有任何固定的有限依赖着色可以这样表达。我们推断涉及的扩展$d$有限类型的尺寸和位移；事实上，任何有限类型的非退化移位也可以区分块因子和有限依赖过程。