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)].