The End of the Double Random Current

Ulrik Thinggaard Hansen; Frederik Ravn Klausen

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arXiv · 2607.07616

The End of the Double Random Current

Ulrik Thinggaard Hansen·Frederik Ravn Klausen

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Abstract

The Ising correlation function can be expressed in terms of connectivity probabilities of a random graph known as the double random current. In 2016, Duminil-Copin asked whether this random graph only has one topological end. One-endedness implies that free and wired random-cluster measures are equal and that any translation-invariant Gibbs measure of the Ising model is a convex combination of the $+$ and $-$ Gibbs measures. In the general setting of transitive amenable graphs, these properties have been established by Raoufi, effectively circumventing the question of one-endedness. In this paper, we prove one-endedness of the double random current on any transitive, amenable one-ended graph and, relying on Raoufi's work, we answer the question of Duminil-Copin in the affirmative in this general setup. The results of this paper do not give a new proof of uniqueness of random-cluster measure or the characterisations of Gibbs states for the Ising model.

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Ulrik Thinggaard Hansen, Frederik Ravn Klausen. 2026-07-08. The End of the Double Random Current. https://arxiv.org/abs/2607.07616

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