Theory of fluctuations in disordered systems

The thesis is devoted to the study of various aspects of disordered and glassy systems. In the first part of the thesis, we have studied the problem of charachterizing the critical dynamical fluctuations of structural glasses at the dynamical transition point. A field theory approach has been developed combined with the replica method and it has been introduced an effective theory that is capable to describe the dynamical heterogeneities at the dynamical transition point. A Ginzburg criterion has been derived to understand the region of validity of the mean field approach. These results are valid for the critical behavior of the dynamics in the beta regime. To understand what happens in the alpha regime we have developed a Boltzmann Pseudodynamics approach to structural glasses that is able to cupture the quasi-equilibrium nature of the glassy dynamics in the long time regime. The third part of the thesis is devoted to the study of the glass and jamming physics of hard spheres in the infinite dimension limit. In this context we show that this model displays a Gardner transition that affects deeply the jamming part of the phase diagram. This means that to describe correctly the jamming properties of hard spheres we need to take into account the full replica symmetry breaking effects. The full replica symmetry breaking formalism for hard sphere systems is completely developed. The last part of the thesis is devoted to study mode coupling dynamics around a quasi-continuous transition.