On eigenvectors of perturbed Toeplitz matrices

We consider Toeplitz matrices perturbed by vanishing noise, i.e. noise of the form $N^{-\gamma} G$ with $\gamma>1/2$ and $G$ having iid zero mean entries of variance $1$; the resulting spectrum concentrates around spectral curves determined by the symbol. For $\gamma>1$, we prove localization of eigenvectors and show they are close to certain pseudo-modes. I will also discuss the case $\gamma\in (1/2,1)$, where a different behavior emerges. Joint work with Anirban Basak and Martin Vogel.

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