# Finitary codings for spatial mixing Markov random fields

## TL;DR

This paper investigates conditions under which spatial mixing Markov random fields can be represented as finitary factors of i.i.d. processes, establishing links between spatial mixing properties and the tail behavior of coding radii.

## Contribution

It proves that weak and strong spatial mixing imply ffiid with specific tail decay rates, extending understanding of phase transition effects on finitary codings.

## Key findings

- Weak spatial mixing implies ffiid with power-law tails.
- Strong spatial mixing implies ffiid with exponential tails.
- Certain critical models satisfy relaxed spatial mixing conditions.

## Abstract

It has been shown by van den Berg and Steif that the sub-critical and critical Ising model on $\mathbb{Z}^d$ is a finitary factor of an i.i.d. process (ffiid), whereas the super-critical model is not. In fact, they showed that the latter is a general phenomenon in that a phase transition presents an obstruction for being ffiid. The question remained whether this is the only such obstruction. We make progress on this, showing that certain spatial mixing conditions (notions of weak dependence on boundary conditions, not to be confused with other notions of mixing in ergodic theory) imply ffiid. Our main result is that weak spatial mixing implies ffiid with power-law tails for the coding radius, and that strong spatial mixing implies ffiid with exponential tails for the coding radius. The weak spatial mixing condition can be relaxed to a condition which is satisfied by some critical two-dimensional models. Using a result of the author, we deduce that strong spatial mixing also implies ffiid with stretched-exponential tails from a finite-valued i.i.d. process. We give several applications to models such as the Potts model, proper colorings, the hard-core model, the Widom--Rowlinson model and the beach model. For instance, for the ferromagnetic $q$-state Potts model on $\mathbb{Z}^d$ at inverse temperature $\beta$, we show that it is ffiid with exponential tails if $\beta$ is sufficiently small, it is ffiid if $\beta < \beta_c(q,d)$, it is not ffiid if $\beta > \beta_c(q,d)$ and, when $d=2$ and $\beta=\beta_c(q,d)$, it is ffiid if and only if $q \le 4$.