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. More precisely, Schramm proved in 2008 that no stationary 1-dependent
3-coloring of the integers exists, and conjectured that no stationary
k-dependent q-coloring exists for any k and q. We disprove this by constructing
a 1-dependent 4-coloring and a 2-dependent 3-coloring, thus resolving the
question for all k and q.
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 Lovasz 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
non-degenerate shift of finite type also distinguishes between block-factors
and finitely dependent processes
