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• For the Potts model on transitive graphs, correlations decay exponentially fast for β < β c .• For the random-cluster model with cluster weight q ≥ 1 on transitive graphs, correlations decay exponentially fast in the subcritical regime and the cluster-density satisfies the meanfield lower bound in the supercritical regime.• For the random-cluster models with cluster weight q ≥ 1 on planar quasi-transitive graphs G,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 [BD12]).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.

We prove that the q-state Potts model and the random-cluster model with cluster weight q > 4 undergo a discontinuous phase transition on the square lattice. More precisely, we show 1. Existence of multiple infinite-volume measures for the critical Potts and random-cluster models, 2. Ordering for the measures with monochromatic (resp. wired) boundary conditions for the critical Potts model (resp. random-cluster model), and 3. Exponential decay of correlations for the measure with free boundary conditions for both the critical Potts and random-cluster models.The proof is based on a rigorous computation of the Perron-Frobenius eigenvalues of the diagonal blocks of the transfer matrix of the six-vertex model, whose ratios are then related to the correlation length of the random-cluster model. As a byproduct, we rigorously compute the correlation lengths of the critical randomcluster and Potts models, and show that they behave as exp(π 2 √ q − 4) as q tends to 4.Section 4: Fourier computations. The study will require certain computations using Fourier decompositions. While these computations are elementary, they may be lengthy, and would break the pace of the proofs. We therefore defer all of them to Section 4.Notation. Most functions hereafter depend on the parameter ∆ = 2−c 2 2 < −1. For ease of notation, we will generally drop the dependency in ∆, and recall it only when it is relevant. We write ∂ i for the partial derivative in the i th coordinate.

We provide a new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. The proof applies to infiniterange models on arbitrary locally finite transitive infinite graphs.For Bernoulli percolation, we prove finiteness of the susceptibility in the subcritical regime β < β c , and the mean-field lower bound P β [0 ←→ ∞] ≥ (β − β c ) β for β > β c . For finite-range models, we also prove that for any β < β c , the probability of an open path from the origin to distance n decays exponentially fast in n.For the Ising model, we prove finiteness of the susceptibility for β < β c , and the mean-field lower boundFor finite-range models, we also prove that the two-point correlation functions decay exponentially fast in the distance for β < β c .The paper is organized in two sections, one devoted to Bernoulli percolation, and one to the Ising model. While both proofs are completely independent, we wish to emphasize the strong analogy between the two strategies.General notation. Let G = (V, E) be a locally finite (vertex-)transitive infinite graph, together with a fixed origin 0 ∈ V . For n ≥ 0, letwhere d(⋅, ⋅) is the graph distance. Consider a set of coupling constants (J x,y ) x,y∈V with J x,y = J y,x ≥ 0 for every x and y in V . We assume that the coupling constants are invariant with respect to some transitively acting group. More precisely, there exists a group Γ of automorphisms acting transitively on V such that J γ(x),γ(y) = J x,y for all γ ∈ Γ. We say that (J x,y ) x,y∈V is finite-range if there exists R > 0 such that J x,y = 0 whenever d(x, y) > R.

We prove that the standard Russo-Seymour-Welsh theory is valid for Voronoi percolation. This implies that at criticality the crossing probabilities for rectangles are bounded by constants depending only on their aspect ratio. This result has many consequences, such as the polynomial decay of the one-arm event at criticality.

We study the phase transition of random radii Poisson Boolean percolation: Around each point of a planar Poisson point process, we draw a disc of random radius, independently for each point. The behavior of this process is well understood when the radii are uniformly bounded from above. In this article, we investigate this process for unbounded (and possibly heavy tailed) radii distributions. Under mild assumptions on the radius distribution, we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter λ c . Moreover,• For λ < λ c , the vacant set has a unique unbounded connected component and we give precise bounds on the one-arm probability for the occupied set, depending on the radius distribution.• At criticality, we establish the box-crossing property, implying that no unbounded component can be found, neither in the occupied nor the vacant sets. We provide a polynomial decay for the probability of the one-arm events, under sharp conditions on the distribution of the radius.• For λ > λ c , the occupied set has a unique unbounded component and we prove that the one-arm probability for the vacant decays exponentially fast.The techniques we develop in this article can be applied to other models such as the Poisson Voronoi and confetti percolation.Mathematics Subject Classification (2010): 60K35, 82B43, 60G55