  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.
