Abstract
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.
Highlights
Central to probability and ergodic theory is the notion of mixing in various forms
A stochastic process is a family of random variables indexed by a metric space, and mixing means that variables at distant locations are approximately c The Author(s) 2016
The strongest and simplest mixing condition is finite dependence, which states that subsets of variables are independent provided they are at least some fixed distance apart
Summary
Central to probability and ergodic theory is the notion of mixing in various forms. A stochastic process is a family of random variables indexed by a metric space, and mixing means that variables at distant locations are approximately c The Author(s) 2016. Theorem 1 and Proposition 2 together provide perhaps the cleanest answer one could hope for to the question raised by Ibragimov and Linnik: Coloring can be done by a stationary 1-dependent process, but not by a block factor. (under the nonlattice condition, it is shown in [22] that there is a finitary factor of an independent and identically distributed process, with tower-function decay of its coding radius, that belongs a.s. to S, and that this decay rate cannot be improved.) Combining this with Corollary 6 provides, as promised, an even more striking answer to the Ibragimov–Linnik question: Any nonlattice shift of finite type on Z that contains no constant sequence serves to distinguish between block factors and stationary finitely dependent processes. We conclude the article with a list of open problems
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