Subcritical sharpness for real-valued spin models

In this paper, we consider a large family of real-valued spin models on general transitive graphs. We show that, in the subcritical regime $\beta <{\beta }_c$, the correlations of the model decay exponentially fast. To prove this result, we consider the random cluster (a.k.a. FK percolation) representation of the model and obtain an inequality that generalises the OSSS inequality to monotonic measures with random connection probabilities, thus extending the inequality of Duminil-Copin, Raoufi, and Tassion. Our results apply in particular to the Blume–Capel model and general $P(\phi )$ models, going beyond the cases of the Ising and ${\phi }^4$ models treated by Aizenman, Barsky, and Fernández.