High dimensional normality of noisy eigenvectors

TL;DR

This paper proves that eigenvectors of large noisy symmetric matrices are jointly asymptotically Gaussian, extending previous eigenvector flow analysis to a multicolor setting and providing explicit distribution computations for various matrix models.

Contribution

It introduces a multicolor eigenvector moment flow analysis for noisy matrices, establishing joint normality and explicit distributions in new settings.

Findings

Eigenvectors converge to a Gaussian distribution in the high-dimensional limit.

The colored eigenvector moment flow exhibits sufficient decay and contraction properties.

Explicit joint eigenvector distributions are derived for Wigner, sparse, and Lévy matrices.

Abstract

We study joint eigenvector distributions for large symmetric matrices in the presence of weak noise. Our main result asserts that every submatrix in the orthogonal matrix of eigenvectors converges to a multidimensional Gaussian distribution. The proof involves analyzing the stochastic eigenstate equation (SEE) which describes the Lie group valued flow of eigenvectors induced by matrix valued Brownian motion. We consider the associated colored eigenvector moment flow defining an SDE on a particle configuration space. This flow extends the eigenvector moment flow first introduced in Bourgade and Yau (2017) to the multicolor setting. However, it is no longer driven by an underlying Markov process on configuration space due to the lack of positivity in the semigroup kernel. Nevertheless, we prove the dynamics admit sufficient averaged decay and contractive properties. This allows us to establish optimal time of relaxation to equilibrium for the colored eigenvector moment flow and prove joint asymptotic normality for eigenvectors. Applications in random matrix theory include the explicit computations of joint eigenvector distributions for general Wigner type matrices and sparse graph models when corresponding eigenvalues lie in the bulk of the spectrum, as well as joint eigenvector distributions for L\'evy matrices when the eigenvectors correspond to small energy levels.