Entropic repulsion of an interface in an external field

We consider an interface above an attractive hard wall in the complete wetting regime, and submitted to the action of an external increasing, convex potential, and study its delocalization as the intensity of this potential vanishes. Our main motivation is the analysis of critical prewetting, which corresponds to the choice of a linear external potential. We also present partial results on critical prewetting in the two dimensional Ising model, as well as a few (weak) results on pathwise estimates for the pure wetting problem for effective interface models

Localization and delocalization of random interfaces

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 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.Comment: Published at http://dx.doi.org/10.1214/154957806000000050 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Critical behavior of the massless free field at the depinning transition

We consider the d-dimensional massless free field localized by a delta-pinning of strength e. We study the asymptotics of the variance of the field, and of the decay-rate of its 2-point function, as e 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 e, for a broad class of d-dimensional massless models

Asymptotics of even-even correlations in the Ising model

We consider finite-range ferromagnetic Ising models on $\mathbb{Z}^d$ in the regime $\beta<\beta_c$. We analyze the behavior of the prefactor to the exponential decay of $\mathrm{Cov}(\sigma_A,\sigma_B)$, for arbitrary finite sets $A$ and $B$ of even cardinality, as the distance between $A$ and $B$ diverges.Comment: Changed section numbering to match the published versio

Self-Attractive Random Walks: The Case of Critical Drifts

Self-attractive random walks undergo a phase transition in terms of the applied drift: If the drift is strong enough, then the walk is ballistic, whereas in the case of small drifts self-attraction wins and the walk is sub-ballistic. We show that, in any dimension at least 2, this transition is of first order. In fact, we prove that the walk is already ballistic at critical drifts, and establish the corresponding LLN and CLT.Comment: Final version sent to the publisher. To appear in Communications in Mathematical Physic

Crossing random walks and stretched polymers at weak disorder

We consider a model of a polymer in $\mathbb{Z}^{d+1}$, 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 $N\to\infty$, when $d\geq3$ 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.Comment: Published in at http://dx.doi.org/10.1214/10-AOP625 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Statistical Mechanics of Stretched Polymers

We describe some recent results concerning the statistical properties of a self-interacting polymer stretched by an external force. We concentrate mainly on the cases of purely attractive or purely repulsive self-interactions, but our results are stable under suitable small perturbations of these pure cases. We provide in particular a precise description of the stretched phase (local limit theorems for the end-point and local observables, invariance principle, microscopic structure). Our results also characterize precisely the (non-trivial, direction-dependent) critical force needed to trigger the collapsed/stretched phase transition in the attractive case. We also describe some recent progress: first, the determination of the order of the phase transition in the attractive case; second, a proof that a semi-directed polymer in quenched random environment is diffusive in dimensions 4 and higher when the temperature is high enough. In addition, we correct an incomplete argument from one of our earlier works

Ballistic Phase of Self-Interacting Random Walks

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 Ornstein-Zernike theory developed in earlier works. 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, Domb-Joyce model, random walks in an annealed random potential, reinforced polymers and weakly reinforced random walks.Comment: One picture and a few annoying typos corrected. Version to be publishe

Potts models with a defect line

We provide a detailed analysis of the correlation length in the direction parallel to a line of modified coupling constants in the ferromagnetic Potts model on $\mathbb{Z}^d$ at temperatures $T>T_c$. We also describe how a line of weakened bonds pins the interface of the Potts model on $\mathbb{Z}^2$ below its critical temperature. These results are obtained by extending the analysis by Friedli, Ioffe and Velenik from Bernoulli percolation to FK-percolation of arbitrary parameter $q>1$.Comment: Final version, as accepted for publication in Communications in Mathematical Physics. (Includes a few improvements in the presentation compared with the previous version.