Noise sensitivity of the top eigenvector of a Wigner matrix

Abstract

We investigate the noise sensitivity of the top eigenvector of a Wigner matrix in the following sense. Let v be the top eigenvector of an \(N\times N\) Wigner matrix. Suppose that k randomly chosen entries of the matrix are resampled, resulting in another realization of the Wigner matrix with top eigenvector \(v^{[k]}\). We prove that, with high probability, when \(k \ll N^{5/3-o(1)}\), then v and \(v^{[k]}\) are almost collinear and when \(k\gg N^{5/3}\), then \(v^{[k]}\) is almost orthogonal to v.