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 model, the random current, double random current and XOR-Ising model.
Acknowledgments
I am grateful to Roland Bauerschmidt, Hugo Duminil-Copin and Aran Raoufi for the discussions on the double random current and XOR-Ising model that we had in 2017 at IHES, Bures-sur-Yvette, and that were the inspiration for this work. I also thank Alexander Glazman and Ron Peled for their very useful comments and suggestions, and Jacques H.H. Perk for bringing to my attention the model of Domany and Riedel [8]
Citation
Marcin Lis. "Spins, percolation and height functions." Electron. J. Probab. 27 1 - 21, 2022. https://doi.org/10.1214/22-EJP761
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