Edge spectra of Gaussian random symmetric matrices with correlated entries

Abstract

We study the largest eigenvalue of a Gaussian random symmetric matrix $X_n$, with zero-mean, unit variance entries satisfying the condition ${\sup }_{(i,j)\ne (i^{\prime },j^{\prime })}|\mathbb{E}[X_{ij}{X}_{i^{\prime }{j}^{\prime }}]|=O(n^{-(1+\epsilon )})$, where $\epsilon >0$. It follows from Catalano et al. (2024) that the empirical spectral distribution of $n^{-1/2}{X}_n$ converges weakly almost surely to the standard semi-circle law. Using a Füredi-Komlós-type high moment analysis, we show that the largest eigenvalue ${\lambda }_1(n^{-1/2}{X}_n)$ of $n^{-1/2}{X}_n$ converges almost surely to $2$. This result is essentially optimal in the sense that one cannot take $\epsilon =0$ and still obtain an almost sure limit of $2$. We also derive Gaussian fluctuation results for the largest eigenvalue in the case where the entries have a common non-zero mean. Let $Y_n=X_n+\frac{\lambda }{\sqrt{n}}11^{\top }$. When $\epsilon \ge 1$ and $\lambda \gg n^{1/4}$, we show that \[ n^{1/2}\bigg(λ_1(n^{-1/2} Y_n) - λ- \frac{1}λ\bigg) \xrightarrow{d} \sqrt{2} Z, \] where $Z$ is a standard Gaussian. On the other hand, when $0<\epsilon <1$, we have $\mathrm{Var}(\frac{1}{n}{\sum }_{i,j}{X}_{ij})=O(n^{1-\epsilon })$. Assuming that $\mathrm{Var}(\frac{1}{n}{\sum }_{i,j}{X}_{ij})={\sigma }^2{n}^{1-\epsilon }(1+o(1))$, if $\lambda \gg n^{\epsilon /4}$, then we have \[ n^{\varepsilon/2}\bigg(λ_1(n^{-1/2} Y_n) - λ- \frac{1}λ\bigg) \xrightarrow{d} σZ. \] While the ranges of $\lambda$ in these fluctuation results are certainly not optimal, a striking aspect is that different scalings are required in the two regimes $0<\epsilon <1$ and $\epsilon \ge 1$.

27 pages, 2 figures; abstract shortened to meet arXiv requirements