Central limit theorems for the real eigenvalues of large Gaussian random matrices

TL;DR

This paper proves a central limit theorem for the sum of polynomial functions of the real eigenvalues of large Gaussian random matrices, showing their fluctuations are normally distributed as matrix size grows.

Contribution

It adapts existing methods to establish a CLT for real eigenvalues of large Gaussian matrices, providing explicit variance formulas and conditions.

Findings

Real eigenvalues exhibit Gaussian fluctuations in large matrices.

Explicit variance formula for polynomial functions of eigenvalues.

Method extends previous approaches to real eigenvalues in Gaussian ensembles.

Abstract

Let $G$ be an $N \times N$ real matrix whose entries are independent identically distributed standard normal random variables $G_{ij} \sim \mathcal{N}(0,1)$. The eigenvalues of such matrices are known to form a two-component system consisting of purely real and complex conjugated points. The purpose of this note is to show that by appropriately adapting the methods of \cite{KPTTZ15}, we can prove a central limit theorem of the following form: if $\lambda_{1},\ldots,\lambda_{N_{\mathbb{R}}}$ are the real eigenvalues of $G$, then for any even polynomial function $P(x)$ and even $N=2n$, we have the convergence in distribution to a normal random variable \begin{equation} \frac{1}{\sqrt{\mathbb{E}(N_{\mathbb{R}})}}\left(\sum_{j=1}^{N_{\mathbb{R}}}P(\lambda_{j})-\mathbb{E}\sum_{j=1}^{N_{\mathbb{R}}}P(\lambda_{j})\right) \to \mathcal{N}(0,\sigma^{2}(P)) \end{equation} as $n \to \infty$, where $\sigma^{2}(P) = \frac{2-\sqrt{2}}{2}\int_{-1}^{1}P(x)^{2}\,dx$.