  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 $q \le 4$.

Probability