I will introduce some interacting particle systems on finite graphs and Cayley graphs of countable groups, and discuss how sofic entropy helps understand them.

More specifically, we consider two notions of statistical equilibrium: an “equilibrium state” maximizes a functional called the pressure while a “Gibbs state” satisfies a local equilibrium condition. On amenable groups (for example, integer lattices) these notions are equivalent, under some assumptions on the interaction. Barbieri and Meyerovitch have recently shown that one direction holds for general sofic groups: equilibrium states are always Gibbs.

I will show that the converse fails in the simplest nontrivial case: the free boundary Ising state on a free group (an infinite regular tree) is Gibbs but not equilibrium. I will also discuss what this says about Gibbs states on finite locally-tree-like graphs: it is well-known that their local statistics are described by some Gibbs state on the infinite tree, but in fact they must locally look like a mixture of equilibrium states. This constraint can be used to compute local limits of finitary Gibbs states for a few interactions.