Non-hermitian random matrices with a variance profile (I): Deterministic equivalents and limiting ESDs

For each n, let An = (σij) be an n × n deterministic matrix and let Xn = (Xij) be an n × n random matrix with i.i.d. centered entries of unit variance. We study the asymptotic behavior of the empirical spectral distribution µYn of the rescaled entry-wise product (Formula presented). For our main result we provide a deterministic sequence of probability measures µn, each described by a family of Master Equations, such that the difference µYn − µn converges weakly in probability to the zero measure. A key feature of our results is to allow some of the entries σij to vanish, provided that the standard deviation profiles An satisfy a certain quantitative irreducibility property. An important step is to obtain quantitative bounds on the solutions to an associate system of Schwinger–Dyson equations, which we accomplish in the general sparse setting using a novel graphical bootstrap argument.

Duke Scholars

Author Nicholas A Cook Mathematics