Outlier eigenvalues for deformed i.i.d. random matrices (Q2828466)

The authors consider a square random matrix of size \(N\) of the form \(A+Y\) where \(A\) is deterministic and \(Y\) has i.i.d. entries with variance \(1/N\). Under mild assumptions, as \(N\) grows, the empirical distribution of the eigenvalues of \(A+Y\) converges weakly to a limit probability measure \(\beta\) on the complex plane. This work is devoted to the study of the outlier eigenvalues, i.e., eigenvalues in the complement of the support of \(\beta\). Even in the simplest cases, a variety of interesting phenomena can occur. The authors give a sufficient condition to guarantee that outliers are stable and provide examples where their fluctuations vary with the particular distribution of the entries of \(Y\) or the Jordan decomposition of \(A\). They also exhibit concrete examples where the outlier eigenvalues converge in distribution to the zeros of a Gaussian analytic function.