Edge spectra of Gaussian random symmetric matrices with correlated entries
Debapratim Banerjee, Soumendu Sundar Mukherjee, Dipranjan Pal
TL;DR
We study the edge of the spectrum for $n\\times n$ symmetric Gaussian matrices with pairwise correlations decaying as $O(n^{-(1+\\varepsilon)})$ and prove that the largest eigenvalue of $n^{-1/2}X_n$ converges to the edge $2$ almost surely. The proof leverages a Füredi–Komlós high-moment analysis with FK-sentence encoding to control trace moments, building on the semicircle law for the empirical spectral distribution. In the spiked model $Y_n = X_n + (\\lambda/\\sqrt{n})\\mathbf{1}\mathbf{1}^\\top$, the top eigenvalue fluctuations are Gaussian with regime-dependent scalings: $\\sqrt{n}$-scaling for $\\varepsilon \ge 1$ and $\\lambda \gg n^{1/4}$, and $n^{\\varepsilon/2}$-scaling when $0<\\varepsilon<1$ with a variance $\\sigma^2$ determined by the mean-sum fluctuations. The authors also develop a universality-type framework via randomised $Q_\\ell$ constructions to extend edge results to a broad class of correlated Gaussian ensembles. Overall, the work extends edge rigidity to correlated Gaussian matrices under weak decay and reveals distinct fluctuation regimes, motivating future exploration of edge universality under dependence structures.
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 Füredi-Komlós-type high moment analysis, we show that the largest eigenvalue $λ_1(n^{-1/2} X_n)$ 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 $Y_n = X_n + \fracλ{\sqrt{n}}\mathbf{1} \mathbf{1}^\top$. When $\varepsilon \ge 1$ and $λ\gg n^{1/4}$, 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 $0 < \varepsilon < 1$, we have $\mathrm{Var}(\frac{1}{n}\sum_{i, j}X_{ij}) = O(n^{1 - \varepsilon})$. Assuming that $\mathrm{Var}(\frac{1}{n}\sum_{i, j} X_{ij}) = σ^2 n^{1 - \varepsilon} (1 + o(1))$, if $λ\gg n^{\varepsilon/4}$, 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 $λ$ 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 $\varepsilon \ge 1$.