Modèles de polymères dans des milieux complexes

In Chapter 2, we describe the results obtained with Frank den Hollander about a model for a copolymer in an emulsion. This model was introduced by den Hollander and Whittington in \cite{dHW06}, and has subsequently being analyzed in details in \cite{dHP07b},\cite{dHP07c}, \cite{dHP07a}. In \cite{dHP13} it was extended to an even more realistic model which enters a new class consisting of those models for a polymer interacting with a random interface. Among the elements of this new class of models we can mention the random walk pinned at another random walk (see \cite{BerTon}, \cite{BS1} and \cite{BS2}) or the famous directed polymer with bulk disorder (see \cite{dH09} for a review). Most models for a copolymer at thermodynamic equilibrium had, up to the introduction of this new class of models, been built with a medium made of a single linear interface separating two solvents. The introduction of a random interface allows us to take into account some more complex media among which are the colloids (milk, ink, polluted water etc...)which are typically stabilized with polymers via the steric/depletion stabilization effect. The particularity of our model for a copolymer in an emulsionis that we managed to provide a variational characterization of its free energy. Such formulas are rarely available for disordered models in Statistical Mechanicsand they turn out to be a powerful tool to analyze the phase diagram of the system.Chapter 3 is dedicated to a completely different phenomenon, that is the collapse transition of a self-interacting partially directed self-avoiding walk. As mentioned above, this is a model for an homopolymer in a repulsive solvent. The model and its non directed counterpart have triggered a fair amount of activity within the physics community (see \cite{BDL09}, \cite{BOP95}, \cite{BGW92} and \cite{cf:Brak}). We studied this model with Philippe Carmona and Gia-Bao Nguyen in \cite{NGP13}, \cite{CNGP13} and \cite{CP}. We analyzed the model with a new probabilistic technic that relies on a random walk representation of the partition function. Our method turns out to be much more explicit and simple than what had been obtained previously with combinatorics techniques. We derive a fine asymptotic for the free energy close to criticality and we provide, for both the collapsed phase and the interior of the extended phase, a complete description of the path under the polymer measure when the system size is large but finite. In Chapter 4, we give a survey of the results obtained in \cite{Petr1}, \cite{Petr} and later in \cite{CP09a}, \cite{CP09b} with Francesco Caravenna and in \cite{CarCarPetr} with Francesco Caravenna and Philippe Carmona. The common point of the models studied in these papers is that they are elements of a wider class of models, i.e., the random walks in a random/deterministic potential. Many celebrated models enters this general framework such as the parabolic Anderson model via the Feynman Kac description of its solution,the pinning of a random walk by one or infinitely many horizontal interfaces, the polymer in a slit interacting with two walls but also the directed polymer in a random environment etc... Most of these models generate some phase transitions between very different regime corresponding to radically differentconformations of the path. We focus on three models that are (1) the discrete parabolic Anderson model, (2) the homogeneous polymer pinned/depinned at infinitely many interfaces and (3) the copolymer randomly pinned at a selective layer between two solvents. We will provide some results concerning the limiting free energy and the critical point for (3), the free energy and the scaling limits of the path in (2) and some strong localization results on the path for (1)