Fluctuations in the spectrum of non-Hermitian i.i.d. matrices.

We consider large non-Hermitian random matrices X with independent identically distributed real or complex entries. In this paper, we review recent results about the eigenvalues of X: (i) universality of local eigenvalue statistics close to the edge of the spectrum of X [Cipolloni et al., "Edge universality for non-Hermitian random matrices," Probab. Theory Relat. Fields 179, 1–28 (2021)], which is the non-Hermitian analog of the celebrated Tracy–Widom universality; (ii) central limit theorem for linear eigenvalue statistics of macroscopic test functions having 2 + ϵ derivatives [Cipolloni et al., "Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices," Commun. Pure Appl. Math. (published online) (2021) and Cipolloni et al., "Fluctuation around the circular law for random matrices with real entries," Electron. J. Probab. 26, 1–61 (2021)]. The main novel ingredients in the proof of these results are local laws for products of two resolvents of the Hermitization of X at two different spectral parameters, coupling of weakly dependent Dyson Brownian motions, and the lower tail estimate for the smallest singular value of X − z in the transitional regime |z| ≈ 1 [Cipolloni et al., "Optimal lower bound on the least singular value of the shifted Ginibre ensemble," Probab. Math. Phys. 1, 101–146 (2020)]. [ABSTRACT FROM AUTHOR]