Edge spectra of Gaussian random symmetric matrices with correlated entries

TL;DR

This paper investigates the asymptotic behavior of the largest eigenvalue of Gaussian symmetric matrices with correlated entries, establishing convergence results and Gaussian fluctuations under various correlation regimes.

Contribution

It provides almost sure convergence of the largest eigenvalue to 2 under certain correlation decay conditions and derives Gaussian fluctuation results for matrices with mean shifts, highlighting different regimes based on correlation decay.

Findings

Largest eigenvalue converges to 2 almost surely for certain correlation decay rates.

Gaussian fluctuation results depend on the correlation decay parameter psilon.

Different scalings are needed for fluctuation results in regimes psilon<1 and psilon1.

Abstract

We study the largest eigenvalue of a Gaussian random symmetric matrix $X_n$, with zero-mean, unit variance entries satisfying the condition $\sup_{(i, j) \ne (i', j')}|\mathbb{E}[X_{ij} X_{i'j'}]| = O(n^{-(1 + \varepsilon)})$, where $\varepsilon > 0$. 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 F\"{u}redi-Koml\'{o}s-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 $\varepsilon = 0$ 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 $Y_n = X_n + \frac{\lambda}{\sqrt{n}}\mathbf{1} \mathbf{1}^\top$. When $\varepsilon \ge 1$ and $\lambda \gg n^{1/4}$, we show that \[ n^{1/2}\bigg(\lambda_1(n^{-1/2} Y_n) - \lambda - \frac{1}{\lambda}\bigg) \xrightarrow{d} \sqrt{2} Z, \] where $Z$ is a standard Gaussian. On the other hand, when $0 < \varepsilon < 1$, we have $\mathrm{Var}(\frac{1}{n}\sum_{i, j}X_{ij}) = O(n^{1 - \varepsilon})$. Assuming that $\mathrm{Var}(\frac{1}{n}\sum_{i, j} X_{ij}) = \sigma^2 n^{1 - \varepsilon} (1 + o(1))$, if $\lambda \gg n^{\varepsilon/4}$, then we have \[ n^{\varepsilon/2}\bigg(\lambda_1(n^{-1/2} Y_n) - \lambda - \frac{1}{\lambda}\bigg) \xrightarrow{d} \sigma Z. \] 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 $0 < \varepsilon < 1$ and $\varepsilon \ge 1$.