Finitely dependent random colorings of bounded degree graphs

Abstract

We prove that every (possibly infinite) graph of degree at most $d$ has a 4-dependent random proper $4^{d(d+1)/2}$-coloring, and one can construct it as a finitary factor of iid. For unimodular transitive (or unimodular random) graphs we construct an automorphism-invariant (respectively, unimodular) 2-dependent coloring by $3^{d(d+1)/2}$ colors. In particular, there exist random proper colorings for $\Z^d$ and for the regular tree that are 2-dependent and automorphism-invariant, or 4-dependent and finitary factor of iid.

Theorems & Definitions (8)

• Theorem 1
• Theorem 2
• Theorem 3: HL, H
• Theorem 4: HHL
• Remark 5
• Lemma 6
• proof
• Remark 7