We provide a detailed analysis of the correlation length in the direction parallel to a line of modified coupling constants in the ferromagnetic Potts model on Z d at temperatures T > Tc. We also describe how a line of weakened bonds pins the interface of the Potts model on Z 2 below its critical temperature. These results are obtained by extending the analysis in [13] from Bernoulli percolation to FK-percolation of arbitrary parameter q ≥ 1.1.1. Correlation length of the Potts model on Z d above T c . Thanks to the self-duality of the 2d Ising model, the problems analyzed in [1,2] admit equivalent reformulations in terms of the inverse correlation length of a 2d Ising model above its critical temperature, in the presence of a line along which the coupling constants are modified. Such an analysis, based on exact computations, was undertaken by McCoy and Perk [20], independently of the previously mentioned works and at the same time. An advantage of this dual version is that it admits immediate generalizations to higher-dimensional lattices. In this section, we investigate this 1 arXiv:1706.09130v4 [math-ph] 18 May 2018 2 SÉBASTIEN OTT AND YVAN VELENIK problem in the more general case of Potts models on Z d . The low temperature setting for the Potts model on Z 2 will be discussed in Section 1.2.GivenLet Ω d q = {1, . . . , q} Z d be the set of configurations of the q-state Potts model on Z d . Given Λ Z d (that is, Λ ⊂ Z d and finite) and J ≥ 0, we associate to ω ∈ Ω d q the energywhere the coupling constants (J i,j ) i,j∈Z d are given byFigure 1.1. The graph of ξ β (J) for the two-dimensional Ising model at β = .75β c , as computed in [20]. As is proved for general Potts models in Theorem 1.2, J c = 1 in this case.(iii) J → ξ β (J) is Lipschitz-continuous and nonincreasing.(iv) There exist c + , c − > 0, depending on β, q and d, such that It follows in particular thatis well defined, for any β < β c , and satisfies ∞ > J c ≥ 1. (See Figure 1.1 for an illustration in the case of the two-dimensional Ising model.)Remark 1.1. The word "longitudinal" above refers to the fact that we consider the correlation length in a direction parallel to the defect line. One could, in a similar fashion, define the transverse correlation length, by replacing e 1 by e 2 in the definition. However, it is not difficult to show that this quantity always coincides with the corresponding quantity in the homogeneous model.Our next result provides information on the value of J c : Theorem 1.2. J c = 1 when d = 2 or d = 3, but J c > 1 when d ≥ 4.When J > J c , more precise information is available.Theorem 1.3. The following properties hold, for any β < β c :(i) J → ξ β (J) is real-analytic and strictly decreasing on (J c , ∞).(ii) When d = 2, there exist c + , c − , > 0, depending on β, q and d, such that, for all J ∈ (J c , J c + ),(iii) When d = 3, there exist c + , c − , > 0, depending on β, q and d, such that, for all J ∈ (J c , J c + ), e −c+/(J−Jc) ≥ ξ β (J c ) − ξ β (J) ≥ e −c−/(J−Jc) .(iv) For all J > J c , there exists C β,J = C β,...

We consider a variety of lattice spin systems (including Ising, Potts and XY models) on $$\mathbb {Z}^d$$ Z d with long-range interactions of the form $$J_x = \psi (x) e^{-|x|}$$ J x = ψ ( x ) e - | x | , where $$\psi (x) = e^{{\mathsf o}(|x|)}$$ ψ ( x ) = e o ( | x | ) and $$|\cdot |$$ | · | is an arbitrary norm. We characterize explicitly the prefactors $$\psi $$ ψ that give rise to a correlation length that is not analytic in the relevant external parameter(s) (inverse temperature $$\beta $$ β , magnetic field $$h$$ h , etc). Our results apply in any dimension. As an interesting particular case, we prove that, in one-dimensional systems, the correlation length is non-analytic whenever $$\psi $$ ψ is summable, in sharp contrast to the well-known analytic behavior of all standard thermodynamic quantities. We also point out that this non-analyticity, when present, also manifests itself in a qualitative change of behavior of the 2-point function. In particular, we relate the lack of analyticity of the correlation length to the failure of the mass gap condition in the Ornstein–Zernike theory of correlations.

We prove that the pressure (or free energy) of the finite range ferromagnetic Ising model on Z d is analytic as a function of both the inverse temperature β and the magnetic field h whenever the model has the exponential weak mixing property. We also prove the exponential weak mixing property whenever h = 0. Together with known results on the regime h = 0, β < β c , this implies both analyticity and weak mixing in all the domain of (β, h) outside of the transition line [β c , ∞) × {0}. The proof of analyticity uses a graphical representation of the Glauber dynamic due to Schonmann and cluster expansion. The proof of weak mixing uses the random cluster representation.

We prove that the correction to exponential decay of the truncated two points function in the homogeneous positive field Ising model is c x −(d−1)/2 . The proof is based on the development in the random current representation of a "modern" Ornstein-Zernike theory, as developed by Campanino, Ioffe and Velenik [7].

We consider nearest-neighbour two-dimensional Potts models, with boundary conditions leading to the presence of an interface along the bottom wall of the box. We show that, after a suitable diffusive scaling, the interface weakly converges to the standard Brownian excursion.