A General Coupling for Ising Models and Beyond

TL;DR

This paper introduces a unified coupling framework for Ising models and related graphical models, enabling new insights into their relationships, phase transitions, and applications to lattice gauge theories.

Contribution

It generalizes the Swendsen-Wang-Edwards-Sokal coupling to a broader class of models, including random currents and loop models, and applies this to prove phase decay equivalences and extend to gauge theories.

Findings

Unified coupling for Ising and graphical models

Proved exponential decay regimes coincide for Potts and related models

Extended coupling framework to lattice gauge theories

Abstract

The couplings between the Ising model and its graphical representations, the random-cluster, random current and loop $\mathrm{O}(1)$ models, are put on common footing through a generalization of the Swendsen-Wang-Edwards-Sokal coupling. A new special case is that the single random current with parameter $2J$ on edges of agreement of XOR-Ising spins has the law of the double random current. The coupling also yields a general mechanism for constructing conditional Bernoulli percolation measures via uniform sampling, providing new perspectives on models such as the arboreal gas, random $d$-regular graphs, and self-avoiding walks. As an application of the coupling for the Loop-Cluster joint model, we prove that the regime of exponential decay of the $q$-state Potts model, the random-cluster representation, and its $q$-flow loop representation coincides on the torus, generalizing the Ising result. Finally, the Loop-Cluster coupling is generalized to lattice gauge theories.