Statistics of extreme and persistent events.

Summary This thesis is devoted to the study of a series of quantities that can be defined and calculated with any stochastic process. In particular, we consider the asymptotic distribution of the extreme value of the process and the statistics of persistent deviations (the probability that the process persists under atypical conditions for long times). First we were interested in the statistics of extremes applied to the study of some disordered systems. In particular, a corrosion model has been studied, looking at the statistics of extremal quantities such as the maximum depth reached by the corrosive agent in the attacked medium and the spontaneous stop time of the corrosion process. Other applications of the asymptotic statistics of extremes to the equilibrium physics of spin glasses are recalled and discussed. The statistics of persistent events is introduced and applied to domain growth dynamics in models such as the diffusion equation, the ising model. It has been recently appreciated that very subtle phenomena are present in these systems. Persistence, defined as the fraction of sites that have always been in the same phase, decreases algebraically with time with a non-trivial exponent. This definition of persistence can be extended by considering the distribution of the local time-averaged magnetization at a given site, m(t), which is a measure of the fraction of time spent in one of the possible phases: the usual persistence exponent then belongs to a continuous family (x) of exponents which describe the persistent deviations of m(t) above a given level x. In particular, minimal models that reproduce this complex phenomenology have been studied and it has been shown how the dynamics of the interfaces between the competing domains during the dynamics can influence the spectra of the generalized persistence exponents of the process.