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 $\varepsilon \ge 1$.

翻译：我们研究了高斯随机对称矩阵 $X_n$ 的最大特征值，其元素满足零均值、单位方差，且满足条件 $\sup_{(i, j) \ne (i', j')}|\mathbb{E}[X_{ij} X_{i'j'}]| = O(n^{-(1 + \varepsilon)})$，其中 $\varepsilon > 0$。根据 Catalano 等人 (2024) 的研究，$n^{-1/2} X_n$ 的经验谱分布几乎必然弱收敛于标准半圆律。利用 F\"{u}redi-Koml\'{o}s 型高阶矩分析，我们证明了 $n^{-1/2} X_n$ 的最大特征值 $\lambda_1(n^{-1/2} X_n)$ 几乎必然收敛于 $2$。这一结果在本质上是最优的，因为不能取 $\varepsilon = 0$ 而仍然得到几乎必然极限 $2$。我们还推导了在元素具有公共非零均值情况下最大特征值的高斯涨落结果。令 $Y_n = X_n + \frac{\lambda}{\sqrt{n}}\mathbf{1} \mathbf{1}^\top$。当 $\varepsilon \ge 1$ 且 $\lambda \gg n^{1/4}$ 时，我们证明 \[ n^{1/2}\bigg(\lambda_1(n^{-1/2} Y_n) - \lambda - \frac{1}{\lambda}\bigg) \xrightarrow{d} \sqrt{2} Z, \] 其中 $Z$ 是标准高斯变量。另一方面，当 $0 < \varepsilon < 1$ 时，我们有 $\mathrm{Var}(\frac{1}{n}\sum_{i, j}X_{ij}) = O(n^{1 - \varepsilon})$。假设 $\mathrm{Var}(\frac{1}{n}\sum_{i, j} X_{ij}) = \sigma^2 n^{1 - \varepsilon} (1 + o(1))$，如果 $\lambda \gg n^{\varepsilon/4}$，那么我们有 \[ n^{\varepsilon/2}\bigg(\lambda_1(n^{-1/2} Y_n) - \lambda - \frac{1}{\lambda}\bigg) \xrightarrow{d} \sigma Z. \] 虽然这些涨落结果中 $\lambda$ 的范围肯定不是最优的，但一个引人注目的方面是，在 $0 < \varepsilon < 1$ 和 $\varepsilon \ge 1$ 两种机制下需要不同的缩放尺度。