Optimal Delocalization for Generalized Wigner Matrices

Abstract

We study the eigenvectors of generalized Wigner matrices with subexponential entries and prove that they delocalize at the optimal rate with overwhelming probability. We also prove high probability delocalization bounds with sharp constants. Our proof uses an analysis of the eigenvector moment flow introduced by Bourgade and Yau (2017) to bound logarithmic moments of eigenvector entries for random matrices with small Gaussian components. We then extend this control to all generalized Wigner matrices by comparison arguments based on a framework of regularized eigenvectors, level repulsion, and the observable employed by Landon, Lopatto, and Marcinek (2018) to compare extremal eigenvalue statistics. Additionally, we prove level repulsion and eigenvalue overcrowding estimates for the entire spectrum, which may be of independent interest.

Thermalisation for Wigner matrices

“…The classical RAGE theorem [15,Section 5.4] for self-adjoint operators H on an infinite dimensional Hilbert space H shows that the Heisenberg time evolution A(t) = e itH Ae −itH of a compact operator A asymptotically vanishes on any state ψ ∈ H in the continuous spectral subspace of H; more precisely ψ, A(t)ψ tends to zero in Cesaro mean for large time t. Since acting on a finite dimensional space, large N × N Wigner matrices W do not have continuous spectrum in a literal sense, but for many physical purposes they still behave as an operator with continuous spectrum; for example their eigenvectors are completely delocalized [20,21,8]. Hence the analogue of the RAGE theorem for Wigner matrices would assert that the matrix elements of A(t) := e itW Ae −itW at any fixed deterministic vectors x, y ∈ C N become very close to their limiting value for large times, i.e.…”

Localization of eigenvectors of non-Hermitian banded noisy Toeplitz matrices

“…As already mentioned in Section 1.1, there are two complementary notions of delocalizations in the random matrix literature. It is shown in [10,47] that both these notions of delocalization hold for Wigner matrices under various assumptions on it entries. There is no reason to believe that these two notions of delocalizations should hold simultaneously in any given setting.…”

Equipartition principle for Wigner matrices

“…Delocalization follows directly from the optimal local law, see e.g. [7], and [2] for an optimal rate. Moreover, the eigenvectors are asymptotically normal, in the sense that for any fixed deterministic vector q ∈ C N the moments of √ N |(q, w)| coincide with those of the modulus of a standard Gaussian [4,10,14].…”