Disordered models: Spin Glasses and Directed Polymers

Disordered systems are among the most difficult and most fascinating problems in statistical mechanics. One speaks of disordered (or complex) system when the dynamics, or the structures that appears within the system, exhibits a rich variety of behaviours, while the microscopic entities the system is made of, and the interactions among these entities, are a priori simple. In this thesis we consider two famous examples of these systems: spin glasses and directed polymers in random environments. In the first part of the thesis we study a variant of the Sherrington-Kirkpatrick (SK) model, the SK model with ferromagnetic interactions. More precisely the Hamiltonian that describes the model is a combination of a SK and Curie-Weiss Hamiltonian. Our aim is to extend the well known results obtained in the SK model, trying to describe this new model in the high temperature region. The main result is that the two key parameters, the magnetization and the overlap, are asymptotically close to constants ? and q, That are the unique solutions of the so-called replica symmetric equations of this model. We then use this result to study the thermodynamical limit of the free energy and the behaviour of the Gibbs measure. In the second part of the thesis we consider two models of directed polymers in random environment. First of all we consider a Brownian polymer in a Gaussian environment, fully determined by its covariance function. In this case it is known that the thermodynamical limit of the free energy exists and it is expected that the polymer is in the strong disorder regime for low temperatures. We give a better estimate of the limit of the free energy in order to quantify how far we are from the weak disorder regime. Then we modify the hypothesis on the covariance of the environment, to determine if one ever leaves the strong disorder regime. After this we consider a continuous time random walk in a white noise potential, making a link between the last result and this new model.