Alain Comtet, Yves Tourigny
Published 2016-01-08Version 1
This is an introduction to the theory of one-dimensional disordered systems and products of random matrices, confined to the 2 by 2 case. The notion of impurity model--- that is, a system in which the interactions are highly localised--- links the two themes and enables their study by elementary mathematical tools. After discussing the spectral theory of some impurity models, we state and illustrate Furstenberg's theorem, which gives sufficient conditions for the exponential growth of a product of independent, identically-distributed matrices.
Comments: This is an extended version of lectures given at the Summer School on Stochastic Processes and Random Matrices, held at the Ecole de Physique, Les Houches, in July 2015. 60 pages and 5 figures
Universality for random matrices and log-gases
Explicit invariant measures for products of random matrices
Regular expansion for the characteristic exponent of a product of $2 \times 2$ random matrices
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