We prove a long-standing conjecture on random-cluster models, namely that the critical point for such models with parameter q 1 on the square lattice is equal to the self-dual pointThis gives a proof that the critical temperature of the q-state Potts model is equal to log(1 + √ q) for all q 2. We further prove that the transition is sharp, meaning that there is exponential decay of correlations in the sub-critical phase. The techniques of this paper are rigorous and valid for all q 1, in contrast to earlier methods valid only for certain given q. The proof extends to the triangular and the hexagonal lattices as well. Mathematics Subject Classification (2000)60K35 · 82B20 (primary); 82B26 · 82B43

Abstract. In r-neighbour bootstrap percolation on a graph G, a (typically random) set A of initially 'infected' vertices spreads by infecting (at each time step) vertices with at least r already-infected neighbours. This process may be viewed as a monotone version of the Glauber dynamics of the Ising model, and has been extensively studied on the ddimensional grid [n] d . The elements of the set A are usually chosen independently, with some density p, and the main question is to determine p c ([n] d , r), the density at which percolation (infection of the entire vertex set) becomes likely.In this paper we prove, for every pair d, r ∈ N with d r 2, thatas n → ∞, for some constant λ(d, r) > 0, and thus prove the existence of a sharp threshold for percolation in any (fixed) number of dimensions. We moreover determine λ(d, r) for every d r 2.

We provide the first mathematical proof that the connective constant of the hexagonal lattice is equal to 2 + √ 2. This value has been derived nonrigorously by B. Nienhuis in 1982, using Coulomb gas approach from theoretical physics. Our proof uses a parafermionic observable for the self-avoiding walk, which satisfies a half of the discrete Cauchy-Riemann relations. Establishing the other half of the relations (which conjecturally holds in the scaling limit) would also imply convergence of the self-avoiding walk to SLE(8/3).

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.

The spontaneous magnetization is proved to vanish continuously at the critical temperature for a class of ferromagnetic Ising spin systems which includes the nearest neighbor ferromagnetic Ising spin model on Z d in d = 3 dimensions. The analysis applies also to higher dimensions, for which the result is already known, and to systems with interactions of power law decay. The proof employs in an essential way an extension of Ising model's random current representation to the model's infinite volume limit. Using it, we relate the continuity of the magnetization to the vanishing of the free boundary condition Gibbs state's Long Range Order parameter. For reflection positive models the resulting criterion for continuity may be established through the infrared bound for all but the borderline lower dimensional cases. The exclusion applies to the one dimensional model with 1 r 2 interaction for which the spontaneous magnetization is known to be discontinuous at T c . x∈Λ hσ x − {x,y}⊂Λ∶x≠y

Abstract. We show how to combine our earlier results to deduce strong convergence of the interfaces in the planar critical Ising model and its random-cluster representation to Schramm's SLE curves with parameter κ = 3 and κ = 16/3 respectively.