Dualities between fermionic theories and the Potts model

We show that a large class of fermionic theories are dual to a $q\to 0$ limit of the Potts model in the presence of a magnetic field. These can be described using a statistical model of random forests on a graph, generalizing the (unrooted) random forest description of the Potts model with only nearest neighbor interactions. We then apply this to find a statistical description of a recently introduced family of $OSp(1\mathrm{\mid }2M)$ invariant field theories that provide a UV completion to sigma models with the same symmetry.