Caltech/UCLA Joint Analysis Seminar

An i.i.d. matrix $X_n$ is an $ntimes n$ random matrix with independent, centered entries of unit variance. The circular law states that in the large $n$ limit the eigenvalues of $X_n$ become uniformly distributed over the origin-centered disk of radius $sqrt{n}$ in the complex plane. In this talk we discuss generalizations of the circular law to centered random matrices $Y_n$ with entries having non-identical variances $sigma_{ij}^2$. Under mild assumptions on the variances we determine the asymptotic spectral distribution for $Y_n$. Key components of the proof are bounds on the smallest singular value for diagonal perturbations of $Y_n$, and analysis of an associated system of Schwinger-Dyson loop equations. Based on joint work with Walid Hachem, Jamal Najim and David Renfrew.