Moderate-to-large deviation asymptotics for real eigenvalues of the elliptic Ginibre matrices
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912727297949696 |
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| author | Byun, Sung-Soo Jalowy, Jonas Lee, Yong-Woo Schehr, Grégory |
| author_facet | Byun, Sung-Soo Jalowy, Jonas Lee, Yong-Woo Schehr, Grégory |
| contents | We study the statistics of the number of real eigenvalues in the elliptic deformation of the real Ginibre ensemble. As the matrix dimension grows, the law of large numbers and the central limit theorem for the number of real eigenvalues are well understood, but the probabilities of rare events remain largely unexplored. Large deviation type results have been obtained only in extreme cases, when either a vanishingly small proportion of eigenvalues are real or almost all eigenvalues are real. Here, in both the strong and weak asymmetry regimes, we derive the probabilities of rare events in the moderate-to-large deviation regime, thereby providing a natural connection between the previously known regime of Gaussian fluctuations and the large deviation regime. Our results are new even for the classical real Ginibre ensemble. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_09191 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Moderate-to-large deviation asymptotics for real eigenvalues of the elliptic Ginibre matrices Byun, Sung-Soo Jalowy, Jonas Lee, Yong-Woo Schehr, Grégory Probability Statistical Mechanics Mathematical Physics We study the statistics of the number of real eigenvalues in the elliptic deformation of the real Ginibre ensemble. As the matrix dimension grows, the law of large numbers and the central limit theorem for the number of real eigenvalues are well understood, but the probabilities of rare events remain largely unexplored. Large deviation type results have been obtained only in extreme cases, when either a vanishingly small proportion of eigenvalues are real or almost all eigenvalues are real. Here, in both the strong and weak asymmetry regimes, we derive the probabilities of rare events in the moderate-to-large deviation regime, thereby providing a natural connection between the previously known regime of Gaussian fluctuations and the large deviation regime. Our results are new even for the classical real Ginibre ensemble. |
| title | Moderate-to-large deviation asymptotics for real eigenvalues of the elliptic Ginibre matrices |
| topic | Probability Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2511.09191 |