Classes of non-Gaussian random matrices: long-range eigenvalue correlations and non-ergodic extended eigenvectors
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908504931958784 |
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| author | Baron, Joseph W. |
| author_facet | Baron, Joseph W. |
| contents | The remarkable universality of the eigenvalue correlation functions is perhaps one of the most salient findings in random matrix theory. Particularly for short-range separations of the eigenvalues, the correlation functions have been shown to be robust to many changes in the random matrix ensemble, and are often well-predicted by results corresponding to Gaussian random matrices in many applications. In this work, we show that, in contrast, the long-range correlations of the eigenvalues of random matrices are more sensitive. Using a path-integral approach, we identify classes of statistical deviations from the Gaussian Orthogonal random matrix Ensemble (GOE) that give rise to long-range correlations. We provide closed-form analytical expressions for the eigenvalue compressibility and two-point correlations, which ordinarily vanish for the GOE, but are non-zero here. These expressions are universal in the limit of small non-Gaussianity. We discuss how these results suggest the presence of non-ergodic eigenvectors, and we verify numerically that the eigenvector component distributions of a wide variety of non-Gaussian ensembles exhibit the associated power-law tails. We also comment on how these findings reveal the need to go beyond simple mean-field theories in disordered systems with non-Gaussian interactions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_14282 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classes of non-Gaussian random matrices: long-range eigenvalue correlations and non-ergodic extended eigenvectors Baron, Joseph W. Disordered Systems and Neural Networks The remarkable universality of the eigenvalue correlation functions is perhaps one of the most salient findings in random matrix theory. Particularly for short-range separations of the eigenvalues, the correlation functions have been shown to be robust to many changes in the random matrix ensemble, and are often well-predicted by results corresponding to Gaussian random matrices in many applications. In this work, we show that, in contrast, the long-range correlations of the eigenvalues of random matrices are more sensitive. Using a path-integral approach, we identify classes of statistical deviations from the Gaussian Orthogonal random matrix Ensemble (GOE) that give rise to long-range correlations. We provide closed-form analytical expressions for the eigenvalue compressibility and two-point correlations, which ordinarily vanish for the GOE, but are non-zero here. These expressions are universal in the limit of small non-Gaussianity. We discuss how these results suggest the presence of non-ergodic eigenvectors, and we verify numerically that the eigenvector component distributions of a wide variety of non-Gaussian ensembles exhibit the associated power-law tails. We also comment on how these findings reveal the need to go beyond simple mean-field theories in disordered systems with non-Gaussian interactions. |
| title | Classes of non-Gaussian random matrices: long-range eigenvalue correlations and non-ergodic extended eigenvectors |
| topic | Disordered Systems and Neural Networks |
| url | https://arxiv.org/abs/2503.14282 |