Eigenvector delocalization in quantum chaos and random matrix theory

Abstract

This thesis consists of two parts concerning delocalization of eigenvectors: the behavior of eigenvectors associated with quantum graphs from classically ergodic interval maps, and a delocalization-localization transition in structured random matrices. In the first part, we prove an analogue of the pointwise Weyl law for eigenvectors of famillies of unitary matrices obtained from quantization of one-dimensional interval maps. This quantization for interval maps was introduced by Pakonski et al. [J. Phys. A 34 9303 (2001)] as a model for quantum chaos on graphs. We allow shrinking spectral windows in the pointwise Weyl law analogue, which allows for a strengthening of the quantum ergodic theorem for these models, and also allows for construction of randomly perturbed quantizations that have approximately Gaussian eigenvectors in the semiclassical limit. The second part is concerned with a localization-delocalization transition for structured random matrices associated with d-regular graphs. This model includes both sparse and non-sparse Gaussian matrices with 1<<d \le N nonzero entries in each row or column, such as random band matrices, as well as various models of interest in computer science and combinatorics. For such matrices, Bandeira and van Handel [Ann. Probab. 44 2479 (2016)] showed that the norm undergoes a phase transition at d~log(N). This transition cannot in general be captured by localization or delocalization of the top eigenvectors, but we show that the transition is captured instead by a localization-delocalization transition of approximate top eigenvectors.