Spins, percolation and height functions

Abstract: To highlight certain similarities in combinatorial representations of several well known two-dimensional models of statistical mechanics, we introduce and study a new family of models which specializes to these cases after a proper tuning of the parameters. To be precise, our model consists of two independent standard Potts models, with possibly different numbers of spins and different coupling constants (the four parameters of the model), defined jointly on a graph embedded in a surface and its dual graph, and conditioned on the event that the primal and dual interfaces between spins of different value do not intersect. We also introduce naturally related height function and bond percolation models, and we discuss their basic properties and mutual relationship. As special cases we recover the standard Potts and random cluster model, the six-vertex model and loop $O(n)$ model, the random current, double random current and XOR-Ising model.