High Dimensional Normality of Noisy Eigenvectors

We study joint eigenvector distributions for large symmetric matrices in the presence of weak noise. Our main result asserts that every submatrix orthogonal matrix eigenvectors converges to a multidimensional Gaussian distribution. The proof involves analyzing stochastic eigenstate equation (SEE) (Bourgade and Yau Comm Math Phys, 2013) which describes Lie group valued flow induced by Brownian motion. consider associated colored moment defining an SDE on particle configuration space. This extends first introduced Bourgade (Comm multicolor setting. However, it is no longer driven underlying Markov process space due lack positivity semigroup kernel. Nevertheless, we prove dynamics admit sufficient averaged decay contractive properties. allows us establish optimal time relaxation equilibrium asymptotic normality eigenvectors. Applications random theory include explicit computations general Wigner type sparse graph models when corresponding eigenvalues lie bulk spectrum, as well Lévy correspond small energy levels.