# Noise sensitivity of the top eigenvector of a Wigner matrix

## TL;DR

This paper studies how the top eigenvector of a Wigner matrix changes when a small or large number of entries are randomly resampled, revealing a phase transition in its sensitivity.

## Contribution

It establishes a precise threshold at which the top eigenvector shifts from being nearly aligned to nearly orthogonal due to entry resampling.

## Key findings

- For k much less than N^{5/3}, eigenvectors remain almost collinear.
- For k much greater than N^{5/3}, eigenvectors become almost orthogonal.
- Identifies a phase transition in eigenvector sensitivity at k ~ N^{5/3}.

## 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 $v$.