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Abstract |
We reconsider the study of the eigenvectors of a random matrix, to better understand the relation between localization and eigenvalue statistics. Traditionally, the requirement of base invariance has lead to the conclusion that invariant models describe only extended (conductive) systems. We show that deviations of the eigenvalue statistics from the Wigner-Dyson universality reflects itself on the eigenvector distribution. In particular, gaps in the eigenvalue density spontaneously break the U(N) symmetry to a smaller one, hence rendering the system not anymore ergodic. Models with log-normal weights, recently considered also in string theory models such as ABJM theories, show a critical eigenvalue distribution which would indicate a critical breaking of the U(N) symmetry, supposedly resulting into a multi-fractal eigenvector statistics. These results pave the way to the exploration of localization problems using random matrices via the study of new classes of observables and potentially to novel, interdisciplinary, applications of matrix models. |
