We develop a fluctuation theory of connectivities for subcritical random cluster models. The theory is based on a comprehensive nonperturbative probabilistic description of long connected clusters in terms of essentially one-dimensional chains of irreducible objects. Statistics of local observables, for example, displacement, over such chains obey classical limit laws, and our construction leads to an effective random walk representation of percolation clusters. The results include a derivation of a sharp Ornstein–Zernike type asymptotic formula for two point functions, a proof of analyticity and strict convexity of inverse correlation length and a proof of an invariance principle for connected clusters under diffusive scaling. In two dimensions duality considerations enable a reformulation of these results for supercritical nearest-neighbor random cluster measures, in particular, for nearest-neighbor Potts models in the phase transition regime. Accordingly, we prove that in two dimensions Potts equilibrium crystal shapes are always analytic and strictly convex and that the interfaces between different phases are always diffusive. Thus, no roughening transition is possible in the whole regime where our results apply. Our results hold under an assumption of exponential decay of finite volume wired connectivities [assumption (1.2) below] in rectangular domains that is conjectured to hold in the whole subcritical regime; the latter is known to be true, in any dimensions, when q=1, q=2, and when q is sufficiently large. In two dimensions assumption (1.2) holds whenever there is an exponential decay of connectivities in the infinite volume measure. By duality, this includes all supercritical nearest-neighbor Potts models with positive surface tension between ordered phases

Abstract. We derive a precise Ornstein-Zernike asymptotic formula for the decay of the two-point function σ 0 σ x β in the general context of finite range Ising type models on Z d . The proof relies in an essential way on the a-priori knowledge of the strict exponential decay of the two-point function and, by the sharp characterization of phase transition due to Aizenman, Barsky and Fernández, goes through in the whole of the high temperature region β < β c . As a byproduct we obtain that for every β < β c , the inverse correlation length ξ β is an analytic and strictly convex function of direction.

We consider the massless field with zero boundary conditions outside D N ≡ D ∩ ‫ޚ(‬ d /N ) (N ∈ ‫ޚ‬ + ), D a suitable subset of ‫ޒ‬ d , i.e. the continuous spin Gibbs measure ‫ސ‬ N on ‫ޒ‬ ‫ޚ‬ d /N with Hamiltonian given by H (ϕ) = x,y:|x−y|=1 V (ϕ(x)−ϕ(y)) and ϕ(x) = 0 for x ∈ D C N . The interaction V is taken to be strictly convex and with bounded second derivative. This is a standard effective model for a (d + 1)-dimensional interface: ϕ represents the height of the interface over the base D N . Due to the choice of scaling of the base, we scale the height with the same factor by setting ξ N = ϕ/N.We study various concentration and relaxation properties of the family of random surfaces {ξ N } and of the induced family of gradient fields {∇ N ξ N } as the discretization step 1/N tends to zero (N → ∞). In particular, we prove a large deviation principle for {ξ N } and show that the corresponding rate function is given by D σ (∇u(x))dx, where σ is the surface tension of the model. This is a multidimensional version of the sample path large deviation principle. We use this result to study the concentration properties of ‫ސ‬ N under the volume constraint, i.e. the constraint that (1/N d ) x∈D N ξ N (x) stays in a neighborhood of a fixed volume v > 0, and the hard-wall constraint, i.e. ξ N (x) ≥ 0 for all x. This is therefore a model for a droplet of volume v lying above a hard wall. We prove that under these constraints the field {ξ N } of rescaled heights concentrates around the solution of a variational problem involving the surface tension, as it would be predicted by the phenomenological theory of phase boundaries. Our principal result, however, asserts local relaxation properties of the gradient field {∇ N ξ N (·)} to the corresponding extremal Gibbs states. Thus, our approach has little in common with traditional large deviation techniques and is closer in spirit to hydrodynamic limit type of arguments. The proofs have both probabilistic and analytic aspects. Essential analytic tools are ‫ތ‬ p estimates for elliptic equations and the theory of Young measures. On the side of probability tools, a central role is played by the Helffer-Sjöstrand (1991): 60K35, 82B24, 35J15 Key words and phrases: Massless fields -Effective interface models -Large deviations -Random walk in Random environment -Wulff variational problem -Winterbottom construction -Linear and nonlinear elliptic PDEs 50 J.-D. Deuschel et al. Mathematics Subject Classification[31] PDE representation for continuous spin systems which we rewrite in terms of random walk in random environment and by recent results of T. Funaki and H. Spohn [25] on the structure of gradient fields.

We explain a unified approach to a study of ballistic phase for a large family of self-interacting random walks with a drift and self-interacting polymers with an external stretching force. The approach is based on a recent version of the OrnsteinZernike theory developed in Campanino et al. (2003, 2004, 2007). It leads to local limit results for various observables (e.g., displacement of the end-point or number of hits of a fixed finite pattern) on paths of n-step walks (polymers) on all possible deviation scales from CLT to LD. The class of models, which display ballistic phase in the "universality class" discussed in the paper, includes self-avoiding walks, DombJoyce model, random walks in an annealed random potential, reinforced polymers and weakly reinforced random walks.

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