Desordre et phenomenes de localisation

TL;DR

This thesis investigates the impact of disorder on the localization of trajectories in random polymer models, including classical and hierarchical models, revealing disorder relevance and superdiffusivity phenomena across dimensions.

Contribution

It demonstrates the relevance of disorder at all temperatures in 1+1 dimensions and the presence of strong disorder at all temperatures in 1+2 dimensions for directed polymers.

Findings

Disorder is relevant at any temperature in 1+1 pinning models.

Strong disorder occurs at all temperatures in 1+2 directed polymers.

Results focus on free energy and superdiffusivity properties.

Abstract

Cette these est consacree a l' etude de differents modeles aleatoires de polymeres. On s'interesse en particulier a l'influence du desordre sur la localisation des trajectoires pour les modeles d'accrochage et pour les polymeres diriges en milieu aleatoire. En plus des modeles classiques dans Zd, nous abordons l' etude de modeles dit hierarchiques, construits sur une suite de reseaux auto-similaires, tres present dans la litterature physique. Les resultats que nous avons obtenus concernent principalement l' energie libre et le phenomene de surdiffusivite. En particulier, nous prouvons: (1) la pertinence du desordre a toute temperature dans pour le modele d' accrochage desordonne en dimension 1 + 1, (2) l' occurence d' un desordre tres fort a toute temperature en dimension 1 + 2 pour les polymeres diriges en milieu aleatoire. This thesis studies models of random directed polymers. We focus on the influence of disorder on localization of the trajectories for pinning model and directed polymers in random environment. In addition to the classical Zd models, we pay a particular attention to so-called hierachical models, buildt on a sequence of self-similar lattices, that are frequently studied in the physics literature. The results we obtain concern mainly free energy and superdiffusivity properties. In particular we present the proof that: (1) disorder is relevant at arbitrary high temperature for pinning models in dimension 1 + 1, (2) very strong disorder holds at all temperature in dimension 1 + 2 for directed polymers in random environment.