Abstract: A process on a graph is a factor of i.i.d. if it can be represented as an automorphism-equivariant function of i.i.d. random variables attached to its vertices, or in other words, there is a “local algorithm” which takes in i.i.d. random variables on the vertices as input, and outputs a sample of the process. Furthermore if for any vertex the “local algorithm” only needs to read the i.i.d. variables from a finite (but random) neighborhood of that vertex, we call the factor finitary.
Previous works showed that the plus/minus state of the Ising model on the Euclidean lattice or any transitive amenable graph is always a factor of i.i.d. and is a finitary factor of i.i.d. if and only if the temperature is at or above criticality. Such questions are not fully understood in the nonamenable setting, even on a regular tree. An important tool in all this analysis is that the Ising model is monotone.
We prove
— The loop 
model (a non-monotone graphical expansion of Ising) is a factor of i.i.d. in various settings (joint with Omer Angel). As a corollary: the gradient of the free state of Ising on planar graphs (even nonamenable) is a factor of i.i.d. This answers a question of Hutchcroft.
— The gradient of the plus/minus/free state of Ising (which is also non-monotone) is a finitary factor of i.i.d. at all temperatures in 
.
We will review some of the standard techniques for proving such results in monotone models and then explain how we build on them in these cases to prove our results.