Yvan Alain Velenik

Ornstein-Zernike theory for finite range Ising models above T c

Abstract. We derive a 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 ℤd. 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<βc. As a byproduct we obtain that for every β<βc, the inverse correlation length ξβ is an analytic and strictly convex function of direction.

Interface, Surface Tension and Reentrant Pinning Transition in the 2D Ising Model

We apply new techniques developed in [PV1] to the study of some surface effects in the 2D Ising model. We examine in particular the pinningdepinning transition. The results are valid for all subcritical temperatures. By duality we obtained new finite size effects on the asymptotic behaviour of the two– point correlation function above the critical temperature. The key–point of the analysis is to obtain good concentration properties of the measure defined on the random lines giving the high–temperature representation of the two–point correlation function, as a consequence of the sharp triangle inequality: let τ̂ (x) be the surface tension of an interface perpendicular to x; then for any x, y τ̂ (x) + τ̂ (y) − τ̂ (x + y) ≥ 1 κ (‖x‖ + ‖y‖ − ‖x + y‖) , where κ is the maximum curvature of the Wulff shape and ‖x‖ the Euclidean norm of x.

2D Models of statistical physics with continuous symmetry: the case of singular interactions

Abstract: We show the absence of continuous symmetry breaking in 2D lattice systems without any smoothness assumptions on the interaction. We treat certain cases of interactions with integrable singularities. We also present cases of singular interactions with continuous symmetry, when the symmetry is broken in the thermodynamic limit.

Random permutations with cycle weights

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.

Fluctuation theory of connectivities for subcritical random cluster models

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.

A note on wetting transition for gradient fields

We prove existence of a wetting transition for two classes of gradient fields which include: (1) The Continuous SOS model in any dimension and (2) The massless Gaussian model in dimension 2. Combined with a recent result proving the absence of such a transition for Gaussian models above 2 dimensions (Bolthausen et al., 2000.) J. Math. Phys. to appear), this shows in particular that absolute-value and quadratic interactions can give rise to completely different behavior.

Crossing random walks and stretched polymers at weak disorder

We consider a model of a polymer in ℤd+1, constrained to join 0 and a hyperplane at distance N. The polymer is subject to a quenched nonnegative random environment. Alternatively, the model describes crossing random walks in a random potential (see Zerner [Ann Appl. Probab. 8 (1998) 246–280] or Chapter 5 of Sznitman [Brownian Motion, Obstacles and Random Media (1998) Springer] for the original Brownian motion formulation). It was recently shown [Ann. Probab. 36 (2008) 1528–1583; Probab. Theory Related Fields 143 (2009) 615–642] that, in such a setting, the quenched and annealed free energies coincide in the limit N → ∞, when d ≥ 3 and the temperature is sufficiently high. We first strengthen this result by proving that, under somewhat weaker assumptions on the distribution of disorder which, in particular, enable a small probability of traps, the ratio of quenched and annealed partition functions actually converges. We then conclude that, in this case, the polymer obeys a diffusive scaling, with the same diffusivity constant as the annealed model.

Critical Behavior of Massless Free Field at Depinning Transition

Abstract: We consider the d-dimensional massless free field localized by a δ-pinning of strength ɛ. 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 ɛ goes to zero, for general Gaussian interactions. Physically speaking, we thus rigorously obtain the critical behavior of the transverse and longitudinal correlation lengths of the corresponding d+ 1-dimensional effective interface model in a non-mean-field regime. We also describe the set of pinned sites at small ɛ, for a broad class of d-dimensional massless models.

Mathematical theory of the wetting phenomenon in the

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.

2

We prove that for a class of massless