On connections between the theory of random operators and the theory of random matrices

Abstract

For several families of selfadjoint ergodic operators, it is proved that, as the parameter that indexes the operators of a family tends to infinity, the integrated density of states converges weakly to the infinite size limit of the normalized counting measure of eigenvalues of certain random matrices. The subsequent informal discussion is devoted to the role of these results as possible indications of the presence of the continuous spectrum for random ergodic operators belonging to the families under consideration, when the indexing parameter values are sufficiently large.