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

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 ${\lambda }_1,\dots ,{\lambda }_{N_{\mathbb{R}}}$ 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(λ_{j})-\mathbb{E}\sum_{j=1}^{N_{\mathbb{R}}}P(λ_{j})\right) \to \mathcal{N}(0,σ^{2}(P)) \end{equation} as $n\to \infty$, where ${\sigma }^2(P)=\frac{2-\sqrt{2}}{2}{\int }_{-1}^{1}P(x{)}^{2}dx$.

References in corpus (4)

• Nonlinear Analogue of the May-Wigner Instability Transition
• Mesoscopic linear statistics of Wigner matrices
• Linear Statistics of Non-Hermitian Matrices Matching the Real or Complex Ginibre Ensemble to Four Moments
• On the logarithm of the characteristic polynomial of the Ginibre ensemble