Spectral deconvolution of matrix models: the additive case

Abstract We implement a complex analytic method to build an estimator of the spectrum of a matrix perturbed by the addition of a random matrix noise in the free probabilistic regime. This method, which has been previously introduced by Arizmendi, Tarrago and Vargas, involves two steps: the first step consists in a fixed point method to compute the Stieltjes transform of the desired distribution in a certain domain, and the second step is a classical deconvolution by a Cauchy distribution, whose parameter depends on the intensity of the noise. This method thus reduces the spectral deconvolution problem to a classical one. We provide explicit bounds for the mean squared error of the first step under the assumption that the distribution of the noise is unitary invariant. In the case where the unknown measure is sparse or close to a distribution with a density with enough smoothness, we prove that the resulting estimator converges to the measure in the $1$-Wasserstein distance at speed $O(1/\sqrt{N})$, where $N$ is the dimension of the matrix.