Edge spectra of Gaussian random symmetric matrices with correlated entries
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2024
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| _version_ | 1866909481624928256 |
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| author | Banerjee, Debapratim Mukherjee, Soumendu Sundar Pal, Dipranjan |
| author_facet | Banerjee, Debapratim Mukherjee, Soumendu Sundar Pal, Dipranjan |
| contents | 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 $\varepsilon \ge 1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_11381 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Edge spectra of Gaussian random symmetric matrices with correlated entries Banerjee, Debapratim Mukherjee, Soumendu Sundar Pal, Dipranjan Probability Mathematical Physics Combinatorics Statistics Theory 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 $\varepsilon \ge 1$. |
| title | Edge spectra of Gaussian random symmetric matrices with correlated entries |
| topic | Probability Mathematical Physics Combinatorics Statistics Theory |
| url | https://arxiv.org/abs/2409.11381 |