Localization for random quasi-one-dimensional models

Abstract

In this paper we review results of Anderson localization for different random families of operators which enter in the framework of random quasi-one-dimensional models. We first recall what is Anderson localization from both physical and mathematical point of views. From the Anderson-Bernoulli conjecture in dimension 2 we justify the introduction of quasi-one-dimensional models. Then we present different types of these models : the Schr{\"o}dinger type in the discrete and continuous cases, the unitary type, the Dirac type and the point-interactions type. In a second part we present tools coming from the study of dynamical systems in dimension one : the transfer matrices formalism, the Lyapunov exponents and the F{\"u}rstenberg group. We then prove a criterion of localization for quasi-one-dimensional models of Schr{\"o}dinger type involving only geometric and algebraic properties of the F{\"u}rstenberg group. Then, in the last two sections, we review results of localization, first for Schr{\"o}dinger type models and then for unitary type models. Each time, we reduce the question of localization to the study of the F{\"u}rstenberg group and show how to use more and more refined algebraic criterions to prove the needed properties of this group. All the presented results for quasi-one-dimensional models of Schr{\"o}dinger type include the case of Bernoulli randomness.