Localization-delocalization phenomena for random interfaces

We consider d-dimensional random surface models which for d=1 are the standard (tied-down) random walks (considered as a random ``string''). In higher dimensions, the one-dimensional (discrete) time parameter of the random walk is replaced by the d-dimensional lattice \Z^d, or a finite subset of it. The random surface is represented by real-valued random variables \phi_i, where i is in \Z^d. A class of natural generalizations of the standard random walk are gradient models whose laws are (formally) expressed as P(d\phi) = 1/Z \exp[-\sum_{|i-j|=1}V(\phi_i-\phi_j)] \prod_i d\phi_i, V:\R -> R^+ convex, and with some growth conditions. Such surfaces have been introduced in theoretical physics as (simplified) models for random interfaces separating different phases. Of particular interest are localization-delocalization phenomena, for instance for a surface interacting with a wall by attracting or repulsive interactions, or both together. Another example are so-called heteropolymers which have a noise-induced interaction. Recently, there had been developments of new probabilistic tools for such problems. Among them are: o Random walk representations of Helffer-Sj\"ostrand type, o Multiscale analysis, o Connections with random trapping problems and large deviations We give a survey of some of these developments

On a nonhierarchical version of the generalized random energy model

We introduce a natural nonhierarchical version of Derrida's generalized random energy model. We prove that, in the thermodynamical limit, the free energy is the same as that of a suitably constructed GREM.Comment: Published at http://dx.doi.org/10.1214/105051605000000665 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Periodic copolymers at selective interfaces: A Large Deviations approach

We analyze a (1+1)-dimension directed random walk model of a polymer dipped in a medium constituted by two immiscible solvents separated by a flat interface. The polymer chain is heterogeneous in the sense that a single monomer may energetically favor one or the other solvent. We focus on the case in which the polymer types are periodically distributed along the chain or, in other words, the polymer is constituted of identical stretches of fixed length. The phenomenon that one wants to analyze is the localization at the interface: energetically favored configurations place most of the monomers in the preferred solvent and this can be done only if the polymer sticks close to the interface. We investigate, by means of large deviations, the energy-entropy competition that may lead, according to the value of the parameters (the strength of the coupling between monomers and solvents and an asymmetry parameter), to localization. We express the free energy of the system in terms of a variational formula that we can solve. We then use the result to analyze the phase diagram.Comment: Published at http://dx.doi.org/10.1214/105051604000000800 in the Annals of Applied Probability (http://www.imstat.org/aap/) 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

Fast decay of covariances under δ−\delta-pinning in the critical and supercritical membrane model

We consider the membrane model, that is the centered Gaussian field on $\mathbb Z^d$ whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a $\delta-$pinning condition, giving a reward of strength $\varepsilon$ for the field to be $0$ at any site of the lattice. In this paper we prove that in dimensions $d\geq 4$ covariances of the pinned field decay at least stretched-exponentially, as opposed to the field without pinning, where the decay is polynomial in $d\geq 5$ and logarithmic in $d=4.$ The proof is based on estimates for certain discrete Sobolev norms, and on a Bernoulli domination result.Comment: 16 pages, 1 figure. Comments are welcom