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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.00675 |
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| _version_ | 1866911187682197504 |
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| author | Bailey, E. Ortiz, S. |
| author_facet | Bailey, E. Ortiz, S. |
| contents | The Keating-Snaith central limit theorem proves that $Λ_N(A)=\log\det(I-A)$, for randomly drawn $A\in \operatorname{U}(N)$, suitably normalised, tends to a complex Gaussian random variable in the large $N$ limit. The deviations of the real and imaginary parts of $Λ_N(A)$, on the scale of a positive $k$th multiple of the variance, are known to be Gaussian but with a multiplicative perturbation in the form of the $2k$th moment coefficient. Here we study the interpolating regime by allowing $k=k(N)$ for both $\operatorname{Re}(Λ_N(A))$ and $\operatorname{Im}(Λ_N(A))$. Additionally our methods apply to the logarithm of the derivative of the characteristic polynomial evaluated at an eigenvalue of $A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_00675 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the interim statistics for compact group characteristic polynomials and their derivatives Bailey, E. Ortiz, S. Mathematical Physics Probability The Keating-Snaith central limit theorem proves that $Λ_N(A)=\log\det(I-A)$, for randomly drawn $A\in \operatorname{U}(N)$, suitably normalised, tends to a complex Gaussian random variable in the large $N$ limit. The deviations of the real and imaginary parts of $Λ_N(A)$, on the scale of a positive $k$th multiple of the variance, are known to be Gaussian but with a multiplicative perturbation in the form of the $2k$th moment coefficient. Here we study the interpolating regime by allowing $k=k(N)$ for both $\operatorname{Re}(Λ_N(A))$ and $\operatorname{Im}(Λ_N(A))$. Additionally our methods apply to the logarithm of the derivative of the characteristic polynomial evaluated at an eigenvalue of $A$. |
| title | On the interim statistics for compact group characteristic polynomials and their derivatives |
| topic | Mathematical Physics Probability |
| url | https://arxiv.org/abs/2510.00675 |