Large deviations from the circular law

Abstract

We prove a full large deviations principle, in the scale N², for the empirical measure of the eigenvalues of an N x N (non self-adjoint) matrix composed of i.i.d. zero mean random variables with variance N^-1. The (good) rate function which governs this rate function possesses as unique minimizer the circular law, providing an alternative proof of convergence to the latter. The techniques are related to recent work by Ben Arous and Guionnet, who treat the self-adjoint case. A crucial role is played by precise determinant computations due to Edelman and to Lehmann and Sommers.