We prove an inequality on decision trees on monotonic measures which generalizes the OSSS inequality on product spaces. As an application, we use this inequality to prove a number of new results on lattice spin models and their random-cluster representations. More precisely, we prove that
1. For the Potts model on transitive graphs, correlations decay exponentially fast for \beta<\beta_c.
2. For the random-cluster model with cluster weight q\geq1 on transitive graphs, correlations decay exponentially fast in the subcritical regime and the cluster-density satisfies the mean-field lower bound in the supercritical regime.
3. For the random-cluster models with cluster weight q\geq1 on planar quasi-transitive graphs G,
\frac{p_c(\mathbb{G})p_c(\mathbb{G}^*)}{(1-p_c(\mathbb{G}))(1-p_c(\mathbb{G}^*))}~=~q.
As a special case, we obtain the value of the critical point for the square, triangular and hexagonal lattices (this provides a short proof of the result of Beffara and Duminil-Copin [Probability Theory and Related Fields, 153(3-4):511--542, 2012]).
These results have many applications for the understanding of the subcritical (respectively disordered) phase of all these models. The techniques developed in this paper have potential to be extended to a wide class of models including the Ashkin-Teller model, continuum percolation models such as Voronoi percolation and Boolean percolation, super-level sets of massive Gaussian Free Field, and random-cluster and Potts model with infinite range interactions.
