Complex Random Matrices have no Real Eigenvalues

Let $\zeta = \xi + i\xi'$ where $\xi, \xi'$ are iid copies of a mean zero, variance one, subgaussian random variable. Let $N_n$ be a $n \times n$ random matrix with entries that are iid copies of $\zeta$. We prove that there exists a $c \in (0,1)$ such that the probability that $N_n$ has any real eigenvalues is less than $c^n$ where $c$ only depends on the subgaussian moment of $\xi$. The bound is optimal up to the value of the constant $c$. The principal component of the proof is an optimal tail bound on the least singular value of matrices of the form $M_n := M + N_n$ where $M$ is a deterministic complex matrix with the condition that $\|M\| \leq K n^{1/2}$ for some constant $K$ depending on the subgaussian moment of $\xi$. For this class of random variables, this result improves on the results of Pan-Zhou and Rudelson-Vershynin. 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 and Rudelson-Vershynin.