Abstract: The 
model is a real-valued spin system with quartic potential. This model has deep connections with the classical Ising model, and both are expected to belong to the same universality class. We construct a random cluster representation for , analogous to that of the Ising model. For this percolation model, we prove that local uniqueness of macroscopic cluster holds throughout the supercritical phase. The corresponding result for the Ising model was proved by Bodineau (2005) and serves as the crucial ingredient in renormalisation arguments used to study fine properties of the supercritical behaviour, such as surface order large deviations, the Wulff construction and exponential decay of truncated correlations. The unboundedness of spins in the model imposes considerable difficulties as compared to the Ising model. This is particularly the case when handling boundary conditions, which we do by relying on the recently constructed random current representation of the model. In the first part, we introduce the spin model and its random cluster representation. We then state our main theorem on local uniqueness, and describe applications to large deviations for the empirical magnetisation and spectral gap decay for the model’s Glauber dynamics. In the second part, we will discuss aspects of the proof, in particular, the key quantitative bounds on crossings that are related to the strict positivity of the (free) surface tension
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