Deterministic equivalence for noisy perturbations

Abstract

We prove a quantitative deterministic equivalence theorem for the logarithmic potentials of deterministic complex $N\times N$ matrices subject to small random perturbations. We show that with probability close to $1$ this log-potential is, up to a small error, determined by the singular values of the unperturbed matrix which are larger than some small $N$-dependent cut-off parameter.