Finitely Dependent Coloring

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 wellseparated 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.

Finitely dependent cycle coloring

“…For (k, q) ∈ {(1, 4), (2, 3)}, the stationary k-dependent q-coloring (X i ) i∈Z constructed in [6] is characterized as follows. Recursively define the function B on [q] n via…”

“…An interesting feature of these colorings is that the supports of proper subsets of the colors can each be expressed as block factors of i.i.d. [6,Theorem 4], even though the coloring as a whole cannot [6,Proposition 2]. Moreover, their descriptions as block factors are remarkably simple and explicit.…”

“…Moreover, their descriptions as block factors are remarkably simple and explicit. However, the proofs used to obtain these descriptions in [6] were in some cases quite involved. Here we introduce a more canonical construction via colorings of the n-cycle, which can be expressed in terms of a necklace insertion process akin to those in [8,9], or in terms of the classical Eden growth model on a 3-regular tree.…”

“…Color symmetry (Theorem 1) of course implies that any other colors may be substituted for 1 and 2 in (i)-(iii) above. The Z case of this theorem was proved in [6,Theorem 4], but the proof was quite complicated and mysterious. We will deduce the Z case from the cycle case, which in turn will be proved by very simple and direct methods.…”

“…We show that this is not the case. [6]. The process (ι(X i )) i∈Z cannot be expressed as a 2-block-factor of an i.i.d.…”