Abstract
For large random matrices $X$ with independent, centered entries but not necessarily identical variances, the eigenvalue density of $XX^*$ is well-approximated by a deterministic measure on $\mathbb{R} $. We show that the density of this measure has only square and cubic-root singularities away from zero. We also extend the bulk local law in [5] to the vicinity of these singularities.
Highlights
The empirical eigenvalue density or density of states of many large random matrices is well-approximated by a deterministic probability measure, the self-consistent density of states
We extend the bulk local law in [5] to the vicinity of these singularities
If X is a p × n random matrix with independent, centered entries of identical variances the limit of the eigenvalue density of the sample covariance matrix XX∗ for large p and n with p/n converging to a constant has been identified by Marchenko and Pastur in [9]
Summary
The empirical eigenvalue density or density of states of many large random matrices is well-approximated by a deterministic probability measure, the self-consistent density of states. For Wigner-type matrices, i.e., Hermitian random matrices with independent (up to the Hermiticity constraint), centered entries, the analogue of (1.1) is a quadratic vector equation (QVE) in the language of [1, 3] In these papers, finite and infinite-dimensional versions of the QVE have been extensively studied to analyze the self-consistent density of states whose Stieltjes transform is the average of the solution to the QVE. The Dyson equation, (1.1), can be transformed into a QVE in the sense of [1] and the spectrum of XX∗ is closely related to that of a Wigner-type matrix in the sense of [2] This is easiest explained on the random matrix level through a special case of the linearization tricks: If X has independent and centered entries the random matrix.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
