We derive precise Ornstein-Zernike asymptotic formula for the decay of the two-point function σ0σx 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,… (More)
We study the distribution of cycle lengths in models of nonuniform random permutations with cycle weights. We identify several regimes. Depending on the weights, the length of typical cycles grows like the total number n of elements, or a fraction of n, or a logarithmic power of n.
We consider the d-dimensional massless free field localized by a δ-pinning of strength ε. We study the asymptotics of the variance of the field (when d = 2), and of the decay-rate of its 2-point function (when d ≥ 2), as ε goes to zero, for general Gaussian interactions. Physically speaking, we thus rigorously obtain the critical behavior of the transverse… (More)
The rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade. We review here the main results which have been obtained, both in two and higher dimensions. In particular, we describe how the phenomenological Wulff and Winterbottom constructions can be derived from the microscopic description provided by the… (More)
We give a mathematical theory of the wetting phenomenon in the 2D Ising model using the formalism of Gibbs states. We treat the grand canonical and canonical ensembles.
We report about concentration results for the measure giving the statistical properties of phase separation lines in the 2D Ising model. The results are sharp and valid for all temperatures below the critical one; they are consequences of a new approach for treating phase separation lines in the 2D Ising model and the following interesting inequality of the… (More)
The probabilistic study of effective interface models has been quite active in recent years, with a particular emphasis on the effect of various external potentials (wall, pinning potential, ...) leading to localiza-tion/delocalization transitions. I review some of the results that have been obtained. In particular, I discuss pinning by a local potential,… (More)
We consider a model of a two-dimensional interface of the SOS type, with finite-range, even, strictly convex, twice continuously differentiable interactions. We prove that, under an arbitrarily weak potential favouring zero-height, the surface has finite mean square heights. We consider the cases of both square well and δ potentials. These results extend… (More)
We develop a fluctuation theory of connectivities for subcritical random cluster models. The theory is based on a comprehensive nonperturbative prob-abilistic 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… (More)