Non-Hermitian Random Matrices with a Variance Profile (II): Properties and Examples

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. In the companion article (Cook et al. in Electron J Probab 23:Paper No. 110, 61, 2018), we considered the empirical spectral distribution μnY of the rescaled entry-wise product Yn=1nAn⊙Xn=(1nσijXij)and provided a deterministic sequence of probability measures μn such that the difference μnY-μn converges weakly in probability to the zero measure. A key feature in Cook et al. (2018) was to allow some of the entries σij to vanish, provided that the standard deviation profiles An satisfy a certain quantitative irreducibility property. In the present article, we provide more information on the sequence (μn) , described by a family of Master Equations. We consider these equations in important special cases such as sampled variance profiles σij2=σ2(in,jn) where (x, y) ↦ σ2(x, y) is a given function on [0 , 1] 2. Associated examples are provided where μnY converges to a genuine limit. We study μn’s behavior at zero. As a consequence, we identify the profiles that yield the circular law. Finally, building upon recent results from Alt et al. (Ann Appl Probab 28(1):148–203, 2018; Ann Inst Henri Poincaré Probab Stat 55(2):661–696, 2019), we prove that, except possibly at the origin, μn admits a positive density on the centered disc of radius ρ(Vn), where Vn=(1nσij2) and ρ(Vn) is its spectral radius.

Duke Scholars

Author Nicholas A Cook Mathematics