One-dependent coloring by finitary factors

TL;DR

This paper demonstrates that a stationary 1-dependent 4-coloring of integers, previously proven to exist, can be explicitly constructed as a finitary factor of an i.i.d. process with power-law tail bounds on the coding radius.

Contribution

It provides an explicit probabilistic construction of the coloring as a finitary factor, extending prior existence results to a constructive framework.

Findings

Explicit finitary factor construction of the coloring.

Power-law tail bounds on the coding radius.

Extension of existence proof to a constructive method.

Abstract

Holroyd and Liggett recently proved the existence of a stationary 1-dependent 4-coloring of the integers, the first stationary k-dependent q-coloring for any k and q. That proof specifies a consistent family of finite-dimensional distributions, but does not yield a probabilistic construction on the whole integer line. Here we prove that the process can be expressed as a finitary factor of an i.i.d. process. The factor is described explicitly, and its coding radius obeys power-law tail bounds.