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 regimes of exponential decay of the $q$-state Potts model, the random-cluster representation, and its $q$-flow loop representation coincide on the torus, generalizing the Ising result. By providing a new and equivalent definition of the plaquette random cluster model, we are able to generalize the loop-cluster coupling to lattice gauge theories.
From: Frederik Ravn Klausen [view email]
[v1]
Thu, 12 Jun 2025 14:47:00 UTC (46 KB)
[v2]
Wed, 5 Aug 2026 12:25:53 UTC (67 KB)
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