具有相关条目高斯随机矩阵的谱边缘
概率论
2025-02-10 v2 数学物理
组合数学
math.MP
统计理论
统计理论
摘要
我们研究了高斯随机对称矩阵 X n X_n X n sup ( i , j ) ≠ ( i ′ , j ′ ) ∣ E [ X i j X i ′ j ′ ] ∣ = O ( n − ( 1 + ε ) ) \sup_{(i, j) \ne (i', j')}|\mathbb{E}[X_{ij} X_{i'j'}]| = O(n^{-(1 + \varepsilon)}) sup ( i , j ) = ( i ′ , j ′ ) ∣ E [ X ij X i ′ j ′ ] ∣ = O ( n − ( 1 + ε ) ) ε > 0 \varepsilon > 0 ε > 0 n − 1 / 2 X n n^{-1/2} X_n n − 1/2 X n n − 1 / 2 X n n^{-1/2} X_n n − 1/2 X n λ 1 ( n − 1 / 2 X n ) \lambda_1(n^{-1/2} X_n) λ 1 ( n − 1/2 X n ) ε = 0 \varepsilon = 0 ε = 0 Y n = X n + λ n 11 ⊤ Y_n = X_n + \frac{\lambda}{\sqrt{n}} \mathbf{1} \mathbf{1}^\top Y n = X n + n λ 1 1 ⊤ ε ≥ 1 \varepsilon \ge 1 ε ≥ 1 λ ≫ n 1 / 4 \lambda \gg n^{1/4} λ ≫ n 1/4 n 1 / 2 ( λ 1 ( n − 1 / 2 Y n ) − λ − 1 λ ) → d 2 Z n^{1/2}\big(\lambda_1(n^{-1/2} Y_n) - \lambda - \frac{1}{\lambda}\big) \xrightarrow{d} \sqrt{2} Z n 1/2 ( λ 1 ( n − 1/2 Y n ) − λ − λ 1 ) d 2 Z Z Z Z 0 < ε < 1 0 < \varepsilon < 1 0 < ε < 1 V a r ( 1 n ∑ i , j X i j ) = O ( n 1 − ε ) \mathrm{Var}(\frac{1}{n}\sum_{i,j} X_{ij}) = O(n^{1 - \varepsilon}) Var ( n 1 ∑ i , j X ij ) = O ( n 1 − ε ) V a r ( 1 n ∑ i , j X i j ) = σ 2 n 1 − ε ( 1 + o ( 1 ) ) \mathrm{Var}(\frac{1}{n}\sum_{i,j} X_{ij}) = \sigma^2 n^{1 - \varepsilon} (1 + o(1)) Var ( n 1 ∑ i , j X ij ) = σ 2 n 1 − ε ( 1 + o ( 1 )) λ ≫ n ε / 4 \lambda \gg n^{\varepsilon/4} λ ≫ n ε /4 n ε / 2 ( λ 1 ( n − 1 / 2 Y n ) − λ − 1 λ ) → d σ Z n^{\varepsilon/2}\big(\lambda_1(n^{-1/2} Y_n) - \lambda - \frac{1}{\lambda}\big) \xrightarrow{d} \sigma Z n ε /2 ( λ 1 ( n − 1/2 Y n ) − λ − λ 1 ) d σ Z λ \lambda λ 0 < ε < 1 0 < \varepsilon < 1 0 < ε < 1 ε ≥ 1 \varepsilon \ge 1 ε ≥ 1
引用
@article{arxiv.2409.11381,
title = {Edge spectra of Gaussian random symmetric matrices with correlated entries},
author = {Debapratim Banerjee and Soumendu Sundar Mukherjee and Dipranjan Pal},
journal= {arXiv preprint arXiv:2409.11381},
year = {2025}
}
备注
27 pages, 2 figures; abstract shortened to meet arXiv requirements