Fluctuations in local quantum unique ergodicity for generalized Wigner matrices

Abstract

We study the eigenvector mass distribution for generalized Wigner matrices on a set of coordinates $I$, where $N^{\epsilon }\le |I|\le N^{1-\epsilon }$, and prove it converges to a Gaussian at every energy level, including the edge, as $N\to \infty$. The key technical input is a four-point decorrelation estimate for eigenvectors of matrices with a large Gaussian component. Its proof is an application of the maximum principle to a new set of moment observables satisfying parabolic evolution equations. Additionally, we prove high-probability Quantum Unique Ergodicity and Quantum Weak Mixing bounds for all eigenvectors and all deterministic sets of entries using a novel bootstrap argument.

44 pages. Minor revisions

References in corpus (3)

• Local semicircle law and complete delocalization for Wigner random matrices
• Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices
• Eigenstate Thermalization Hypothesis for Wigner Matrices

Cited by in corpus (4)

• Entanglement Entropy of Non-Hermitian Eigenstates and the Ginibre Ensemble
• Non-Hermitian Hamiltonians Violate the Eigenstate Thermalization Hypothesis
• Universal characteristics of deep neural network loss surfaces from random matrix theory
• Eigenvectors of the square grid plus GUE