Large Deviations from the Circular Law

We prove a full large deviations principle in the scale N for the empirical measure of the eigenvalues of an N N non self adjoint matrix composed of i i d zero mean random variables with variance N The good rate function which governs this rate func tion possesses as unique minimizer the circular law providing an alter native 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 Introduction Let X fXN ij g be an N N matrix whose entries are independent cen tered normal random variables of variance N We denote the complex eigenvalues of XN by Zi i N and form the empirical measure N N X i Zi The law of N is denoted by Q N The celebrated circular law c f e g Edelman A Girko V L Mehta M L states that  converges in distribution to the uniform law U on the disc D fZ jZj g Our goal in this paper is to study the corresponding large deviations We follow a similar study which was carried out in Ben Arous G Guionnet A for the case of self adjoint matrices In that case the large deviations uctuation have speed N and a rate function related to Voiculescu s non commutative entropy More precisely if Y N fY N ij g is an N N self adjoint matrix with inde pendent for j i centered normal random variables of variance N variance N if i j and eigenvalues yi and if N N PN i yi then satis es the large deviation principle with speed N and rate function IR Z R x dx R log where R Z