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

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60K35 Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
- 82B41 Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]
- 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.)

**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*