  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 $\mathbb{G}$,
  $$\frac{p_c(\mathbb{G})p_c(\mathbb{G}^*)}{(1-p_c(\mathbb{G}))(1-p_c(\mathbb{G}^*))}~=~q.$$
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.
