Spins, percolation and height functions

Abstract

To highlight certain similarities in graphical 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 coupled height function and bond percolation models, and we discuss fundamental properties of the resulting joint coupling.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.

Phase Diagram of the Ashkin–Teller Model

“…In this section, we describe the connection between the self-dual Ashkin–Teller model on a domain in and the spin representation of the six-vertex model on the corresponding even domain in . This relation has first been noticed in [ Fan72a ] comparing their critical properties, it was made explicit in [ Fan72b , Weg72 ] (see also [ HDJS13 ]), and was upgraded to a coupling in [ GP23 , Lis22 ] (we note that [ Lis22 ] treats a more general case of two interacting Potts models). We consider two types of boundary conditions that will play an important role in proving Proposition 1.1 .…”

Finitary codings for gradient models and a new graphical representation for the six‐vertex model

“…This allows us to apply our general result. A short time after the first draft of this paper appeared online, it came to our attention that Marcin Lis [36] had independently obtained a similar representation for a wider class of models.…”