Diluted models in statistical mechanics: Out of equilibrium dynamics and optimization algorithms

This thesis is devoted to the study of dynamical properties of diluted models. These are mean field statistical mechanics systems, but with finite local connectivity. Among other reasons, the interest for these models arises from their deep relationship with combinatorial optimization problems, random $K$-satisfiability for instance. Several analytical descriptions of their out of equilibrium regime are described. This regime can be due to long relaxation times in glassy phases, lack of detailed balance condition for optimization algorithms, or transient relaxation from an arbitrary initial condition for ferromagnets. In the course of these studies some attention will also be given to random matrix theory, and to a generalization of fluctuation-dissipation theorem for $n$-times functions.