Partial linear eigenvalue statistics for non-Hermitian random matrices

TL;DR

This paper investigates the fluctuations of partial sums of eigenvalues in non-Hermitian random matrices when some eigenvalues are removed, revealing non-Gaussian limits and providing convergence rates to the circular law.

Contribution

It extends the understanding of eigenvalue statistics by analyzing partial sums with eigenvalue removal, identifying their limiting distributions, and establishing convergence rates.

Findings

Limiting distribution can be non-Gaussian when eigenvalues are removed.

Results hold for fixed and growing number of removed eigenvalues.

Provides a rate of convergence to the circular law in Wasserstein distance.

Abstract

For an $n \times n$ independent-entry random matrix $X_n$ with eigenvalues $\lambda_1, \ldots, \lambda_n$, the seminal work of Rider and Silverstein asserts that the fluctuations of the linear eigenvalue statistics $\sum_{i=1}^n f(\lambda_i)$ converge to a Gaussian distribution for sufficiently nice test functions $f$. We study the fluctuations of $\sum_{i=1}^{n-K} f(\lambda_i)$, where $K$ randomly chosen eigenvalues have been removed from the sum. In this case, we identify the limiting distribution and show that it need not be Gaussian. Our results hold for the case when $K$ is fixed as well as the case when $K$ tends to infinity with $n$. The proof utilizes the predicted locations of the eigenvalues introduced by E. Meckes and M. Meckes. As a consequence of our methods, we obtain a rate of convergence for the empirical spectral distribution of $X_n$ to the circular law in Wasserstein distance, which may be of independent interest.