Interior eigenvalue density of Jordan matrices with random perturbations

We study the eigenvalue distribution of a large Jordan block subject to a small random Gaussian perturbation. A result by E.B. Davies and M. Hager shows that as the dimension of the matrix gets large, with probability close to $1$, most of the eigenvalues are close to a circle. We study the expected eigenvalue density of the perturbed Jordan block in the interior of that circle and give a precise asymptotic description.

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