The End of the Double Random Current

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.