Limiting Statistics of the Largest and Smallest Eigenvalues in the Correlated Wishart Model

TL;DR

This paper develops a unified approach to analyze the eigenvalue statistics of correlated Wishart matrices, revealing that Tracy-Widom distributions can describe the largest and smallest eigenvalues even in highly correlated settings.

Contribution

It introduces a new method mapping invariant quantities to Hermitian matrix models, applicable to real, complex, and quaternion cases, to study eigenvalue distributions.

Findings

Tracy-Widom distribution applies to extreme eigenvalues in correlated Wishart models.

The approach unifies analysis across real, complex, and quaternion cases.

Eigenvalue statistics depend on a specific scaling of empirical eigenvalues.

Abstract

The correlated Wishart model provides a standard tool for the analysis of correlations in a rich variety of systems. Although much is known for complex correlation matrices, the empirically much more important real case still poses substantial challenges. We put forward a new approach, which maps arbitrary statistical quantities, depending on invariants only, to invariant Hermitian matrix models. For completeness we also include the quaternion case and deal with all three cases in a unified way. As an important application, we study the statistics of the largest eigenvalue and its limiting distributions in the correlated Wishart model, because they help to estimate the behavior of large complex systems. We show that even for fully correlated Wishart ensembles, the Tracy-Widom distribution can be the limiting distribution of the largest as well as the smallest eigenvalue, provided that a certain scaling of the empirical eigenvalues holds.