Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Estimates on path delocalization for copolymers at selective interfaces


Code(s) de Classification MSC:

Résumé: Starting from the simple symmetric random walk $\{ S_n \}_n$, we introduce a new process whose path measure is weighted by a factor $\exp\left( \lambda \sum_{n=1}^N \left(\omega_n +h \right ) \sign \left( S_n\right)\right)$, with $\lambda , h \ge 0$, $\{ \omega _n \}_n $ a typical realization of an IID process and $N$ a positive integer. We are looking for results in the large $N$ limit. This factor favors $S_n>0$ if $\omega_n >0$ and $S_n<0$ if $\omega_n <0$. The process can be interpreted as a model for a random% heterogeneous polymer in the proximity of an% interface separating two selective solvents. It has been shown that this model undergoes a (de)localization transition: more precisely there exists a continuous increasing function $\lambda \longmapsto h_c(\lambda)$ such that if $h< h_c(\lambda)$ then the model is localized while it is delocalized if $h\ge h_c(\lambda)$. However, localization and delocalization were not given in terms of path properties, but in a free energy sense. Later on it has been shown that free energy localization does indeed correspond to a (strong) form of path localization. On the other hand, only weak results on the delocalized regime have been known so far. We present a method, based on concentration bounds on {\sl suitably restricted} partition functions, that yields much stronger results on the path behavior in the interior of the delocalized region, that is for $h> h_c (\lambda)$. In particular we prove that, in a suitable sense, one cannot expect more than $O(\log N)$ visits of the walk to the lower half plane. The previously known bound was $o(N)$. Stronger $O(1)$--type results are obtained deep inside the delocalized region. The same approach is also helpful for a different type of question: we prove in fact that the limit as $\lambda$ tends to zero of $h_c(\lambda) / \lambda$ exists and it is independent of the law of $\omega _1$, at least when the random variable $\omega_1$ is bounded or it is Gaussian. This is achieved by interpolating between this class of variables and the particular case of $\omega_1$ taking values $\pm 1$ with probability $1/2$.

Mots Clés: Copolymers ; Directed Polymers ; Delocalization Transition ; Concentration Inequalities ; Interpolation Techniques

Date: 2004-09-17

Prépublication numéro: PMA-933