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.

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We give a proof of the existence of a (~2 even solution of Feigenbaum's functional equationwhere g is a map of [-1, 1] into itself. It extends to a real analytic function over IK * Permanent address:

Let H=-Δ + V on / 2 (Z), where V(x) 9 xeZ, are ϋ.d.r.v.'s with common probability distribution v. Let h(t) = §e~i tv dv(v) and let k(E) be the integrated density of states. It is proven: (i) If h is n-times differentiable with h (j \t) = 0((l + \t\Γ*) for some α>0, 7 = 0, l,...,n, then k(E) is a C n function. In particular, if v has compact support and h(t) = 0((l H-1 ί |)~α) with α > 0, then k(E) is C°°. This allows v to be singular continuous, (ii) lfh(t) = 0(e~a ]tl ) for some α > 0 then k(E) is analytic in a strip about the real axis.The proof uses the supersymmetric replica trick to rewrite the averaged Green's function as a two-point function of a one-dimensional supersymmetric field theory which is studied by the transfer matrix method.