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.
On the Description of Macroscopic SystemsGases, liquids and solids are the most familiar examples of large physical systems of particles encountered in nature. In the first part of this introduction, for the sake of simplicity and concreteness, we will mainly consider the system of a gas contained in a vessel.
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.
The study of effective interface models has been quite active recently, with a particular emphasis on the effect of various external potentials (wall, pinning potential,...) leading to localization/delocalization transitions. I review some of the results that have been obtained. In particular, I discuss pinning by a local potential, entropic repulsion and the (pre)wetting transition, both for models with continuous and discrete heights. This text is based on lecture notes for a mini-course given during the workshop "Topics in Random Interfaces and Directed Polymers" held in Leipzig, September 12-17 2005
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