The Circular Law for Random Matrices with Intra-row Dependence

TL;DR

This paper extends the circular law to certain random matrices with dependent rows, providing new geometric conditions under which the spectral distribution converges, broadening the applicability of classical random matrix results.

Contribution

It introduces geometric conditions on row distributions that allow the circular law to hold for matrices with intra-row dependence, extending classical results.

Findings

Extended the circular law to matrices with dependent rows

Provided geometric conditions ensuring spectral distribution convergence

Generalized the Marčenko-Pastur theorem for this setting

Abstract

We consider the problem of determining the limiting spectral distribution for random matrices whose row distributions are permitted to have limited dependence. We assume mild moment conditions and give an extension of the Mar\v{c}enko-Pastur theorem for this context. The main new feature here are geometric conditions on the distributions which allow us to extend the circular law to this setting.