Complex random matrices have no real eigenvalues

Abstract

Let [Formula: see text] where [Formula: see text] are iid copies of a mean zero, variance one, subgaussian random variable. Let [Formula: see text] be an [Formula: see text] random matrix with entries that are iid copies of [Formula: see text]. We prove that there exists a [Formula: see text] such that the probability that [Formula: see text] has any real eigenvalues is less than [Formula: see text] where [Formula: see text] only depends on the subgaussian moment of [Formula: see text]. The bound is optimal up to the value of the constant [Formula: see text]. The principal component of the proof is an optimal tail bound on the least singular value of matrices of the form [Formula: see text] where [Formula: see text] is a deterministic complex matrix with the condition that [Formula: see text] for some constant [Formula: see text] depending on the subgaussian moment of [Formula: see text]. For this class of random variables, this result improves on the results of Pan–Zhou [Circular law, extreme singular values and potential theory, J. Multivariate Anal. 101(3) (2010) 645–656] and Rudelson–Vershynin [The Littlewood–Offord problem and invertibility of random matrices, Adv. Math. 218(2) (2008) 600–633]. In the proof of the tail bound, we develop an optimal small-ball probability bound for complex random variables that generalizes the Littlewood–Offord theory developed by Tao–Vu [From the Littlewood–Offord problem to the circular law: Universality of the spectral distribution of random matrices, Bull. Amer. Math. Soc.[Formula: see text]N.S.[Formula: see text] 46(3) (2009) 377–396; Inverse Littlewood–Offord theorems and the condition number of random discrete matrices, Ann. of Math.[Formula: see text] 169(2) (2009) 595–632] and Rudelson–Vershynin [The Littlewood–Offord problem and invertibility of random matrices, Adv. Math. 218(2) (2008) 600–633; Smallest singular value of a random rectangular matrix, Comm. Pure Appl. Math. 62(12) (2009) 1707–1739].