Supercritical sharpness for the random cluster representation of real-valued spin models

We study the supercritical regime $\beta >{\beta }_c$ of a large family of real-valued spin models on ${\mathbb{Z}}^d$, including the Blume-Capel and $P(\phi )$ models. We consider their random cluster representations and prove that they are well-behaved, in the sense that local uniqueness of macroscopic clusters occurs with high probability, uniformly in the boundary conditions. This implies, among other things, a surface-order exponential bound for the (lower) large deviations of the empirical magnetisation. These results were previously known only in the cases of the Ising and ${\phi }^4$ models.