Localization for Random Band Matrices and Quantum Unique Ergodicity for Wigner Matrices

This thesis is a compilation of two papers that are essentially independent. In Chapter One, we first introduce the development of Anderson localization and its relevance to the study of certain random band matrices. The presentation of the background closely follows [Stolz, 2011,Rojas-Molina, 2018]. Then we show the eigenvectors of a random Gaussian band matrix are localized when the band width is less than the 1/4 power of the matrix size. This project is a joint work with Charles Smart. Our approach involves showing exponential decay through logarithmic fluctuations. The argument is adapted from Schenker's proof of the 1/8 exponent in [Schenker, 2009] by analyzing the marginal distribution of a scalar degree of freedom. In Chapter Two, we analyze the fluctuations in Quantum Unique Ergodicity (QUE) at the spectral edge. This project is a joint work with Lucas Benigni, Patrick Lopatto and Xiaoyu Xie. We study the eigenvector mass distribution of an N × N Wigner matrix on a set of coordinates I satisfying |I| ⩾ cN for some constant c > 0. For eigenvectors corresponding to eigenvalues at the spectral edge, we show that the sum of the mass on these coordinates converges to a Gaussian in the N → ∞ limit, after a suitable rescaling and centering. The proof proceeds by a two moment matching argument. We directly compare edge eigenvector observables of an arbitrary Wigner matrix to those of a Gaussian matrix, which may be computed explicitly.