Random polymers

January 25, 2006, IHP, Amphi Hermite

Moderator: Herbert Spohn (München)

Ken Alexander (Los Angeles): The effect of disorder on the polymer depinning transition

We consider a polymer, with monomer locations modeled by the trajectory of a Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0. We assume the probability of an excursion of length $n$ from 0, in the absence of the potential, decays like $n^{-c}$ for some $c>1$. Disorder is introduced by, having the interaction vary from one monomer to another, as a constant $u$ plus i.i.d. mean-0 randomness. There is a critical value of $u$ above which the polymer is pinned, placing a positive fraction, called the contact fraction, of its monomers at 0 with high probability. We obtain upper and lower bounds for the contact fraction, at high temperature, which show that disorder significantly alters the transition but only in a small neighborhood of the critical point, when $c>3/2$. Our results are consistent with predictions in the physics literature that the effect of disorder is quite different from the case $c<3/2$.

Francis Comets (Paris): Directed polymers in random media; majorizing cascades

We review discrete models of directed polymers in random media, with special emphasis on delocalization/localization. Then, we report on a recent work with Vincent Vargas, showing that strong localization of the polymer is the rule in (transverse) dimension 1. Upper bounds on the free energy are given by the so-called generalized multiplicative cascades, a model known from the statistical theory of turbulence.

Giambattista Giacomin (Paris): Depinning transitions for directed polymers in presence of quenched disorder

We present recent results on disordered directed polymer models displaying a localization/delocalization transition. This class includes $(1+1)$--dimensional interface wetting models, random copolymers at a selective interface, disordered Poland--Scheraga models of DNA denaturation and other $(1+d)$--dimensional polymers in interaction with columnar defects. We will present and discuss in particular:
- a free energy estimate that shows that, as soon as disorder is present, the transition is at least of second order;
- estimates on the polymer path behavior in the delocalized regime.

Cecile Monthus (Saclay): Random polymers and delocalization transitions

According to recent progresses in the finite size scaling theory of disordered systems, thermodynamic observables are not self-averaging at critical points whenever disorder is relevant. This lack of self-averageness at criticality is directly related to the scaling properties of the probability distribution of pseudo-critical temperatures $T_c(i,L)$ over the ensemble of samples $(i)$ of size $L$. This framework is very useful to characterize various delocalization transitions involving random polymers :
(i) wetting transition in dimension $1+1$,
(ii) Poland-Scheraga model of DNA denaturation
(iii) the selective interface model.

Stu G. Whittington (Toronto): Coloured self-avoiding walks: a model of copolymer localization

Random copolymers with hydrophyllic and lyophyllic monomers can localize at the interface between two immiscible liquids. This localized phase is stable at low temperatures but, at higher temperatures, entropy dominates and the copolymer delocalizes into one of the two bulk phases. The localization-delocalization phase transition has been studied for several different configurational models. When the polymer is treated as a directed or random walk detailed information about the phase transition can be obtained. The treatment of the self-avoiding walk model is more difficult but the qualitative behaviour is quite well-understood. We shall review the rigorous results known for the self-avoiding walk model and highlight some open problems.

Kay Wiese (ENS, Paris): Elastic manifolds in disorder

I will discuss what renormalization group methods, more precisely Functional Renormalization can teach us about Elastic Manifolds in Disordered Media.