Large deviations for the extremal eigenvalues of Ginibre ensembles

Abstract

We establish large deviation principles for the extremal eigenvalues of the Ginibre ensembles with good rate functions. In contrast to the typical estimates for the extremal eigenvalues, the large deviations for the real Ginibre ensemble come from the eigenvalues lying on the real line. Moreover, we also derive deviation estimates for the second leading term in the asymptotic expansion of the extremal eigenvalues. These polynomially small deviation estimates are universal for any i.i.d. matrices under a mild moment condition.