On constrained annealed bounds for pinning and wetting models

The free energy of quenched disordered systems is bounded above by the free energy of the corresponding annealed system. This bound may be improved by applying the annealing procedure, which is just Jensen inequality, after having modified the Hamiltonian in a way that the quenched expressions are left unchanged. This procedure is often viewed as a partial annealing or as a constrained annealing, in the sense that the term that is added may be interpreted as a Lagrange multiplier on the disorder variables. In this note we point out that, for a family of models, some of which have attracted much attention, the multipliers of the form of empirical averages of local functions cannot improve on the basic annealed bound from the viewpoint of characterizing the phase diagram. This class of multipliers is the one that is suitable for computations and it is often believed that in this class one can approximate arbitrarily well the quenched free energy.Comment: 10 page

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

On the irrelevant disorder regime of pinning models

Recent results have lead to substantial progress in understanding the role of disorder in the (de)localization transition of polymer pinning models. Notably, there is an understanding of the crucial issue of disorder relevance and irrelevance that is now rigorous. In this work, we exploit interpolation and replica coupling methods to obtain sharper results on the irrelevant disorder regime of pinning models. In particular, in this regime, we compute the first order term in the expansion of the free energy close to criticality and this term coincides with the first order of the formal expansion obtained by field theory methods. We also show that the quenched and quenched averaged correlation length exponents coincide, while, in general, they are expected to be different. Interpolation and replica coupling methods in this class of models naturally lead to studying the behavior of the intersection of certain renewal sequences and one of the main tools in this work is precisely renewal theory and the study of these intersection renewals.Comment: Published in at http://dx.doi.org/10.1214/09-AOP454 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Sharp asymptotic behavior for wetting models in (1+1)-dimension

We consider continuous and discrete (1+1)-dimensional wetting models which undergo a localization/delocalization phase transition. Using a simple approach based on Renewal Theory we determine the precise asymptotic behavior of the partition function, from which we obtain the scaling limits of the models and an explicit construction of the infinite volume measure (thermodynamic limit) in all regimes, including the critical one.Comment: 14 pages, 1 figur

Infinite volume limits of polymer chains with periodic charges

The aim of this paper is twofold: - To give an elementary and self-contained proof of an explicit formula for the free energy for a general class of polymer chains interacting with an environment through periodic potentials. This generalizes a result in [Bolthausen and Giacomin, AAP 2005] in which the formula is derived by using the Donsker-Varadhan Large Deviations theory for Markov chains. We exploit instead tools from renewal theory. - To identify the infinite volume limits of the system. In particular, in the different regimes we encounter transient, null recurrent and positive recurrent processes (which correspond to delocalized, critical and localized behaviors of the trajectories). This is done by exploiting the sharp estimates on the partition function of the system obtained by the renewal theory approach. The precise characterization of the infinite volume limits of the system exposes a non-uniqueness problem. We will however explain in detail how this (at first) surprising phenomenon is instead due to the presence of a first-order phase transition.Comment: 27 pages, 2 figure