Regularization of non-normal matrices by Gaussian noise

We consider the regularization of matrices $M^N$ written in Jordan form by additive Gaussian noise $N^{-γ}G^N$, where $G^N$ is a matrix of i.i.d. standard Gaussians and $γ>1/2$ so that the operator norm of the additive noise tends to $0$ with $N$. Under mild conditions on the structure of $M^N$ we evaluate the limit of the empirical measure of eigenvalues of $M^N+N^{-γ} G^N$ and show that it depends on $γ$, in contrast with the case of a single Jordan block.

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