aiqbench analysis
This research introduces a novel numerical method to characterize system symmetry structures using higher-order spacing statistics in superposed random matrix spectra, offering a tool that avoids the need for spectral desymmetrization. By validating conjectures against the quantum chaotic kicked top model, the study provides a robust framework for distinguishing between different symmetry ensembles (COE/GOE) across varying Hilbert space dimensions. The work advances the theoretical toolkit for analyzing spectral fluctuations in complex quantum systems, with potential applications in identifying underlying symmetry classes from raw spectral data.
Higher-order spacings in the superposed spectra of random matrices with comparison to spacing ratios and application to complex systems
arXiv:2510.00503v2 Announce Type: replace-cross Abstract: Higher-order spacing statistics in the $m$ superposed spectra of circular random matrices of the same class are studied numerically. We conjecture that for given $m$ (or order $k$) and $\beta$, the sequence of modified Dyson index $\beta'(k)$ (or $\beta'(m)$) obtained using the sum of absolute differences between the cumulative distribution functions method (denoted as $D(\beta')$) is unique. Also, for a given $k$, the distribution tends to the corresponding $k$-th order Poisson statistics in the limit $m\rightarrow \infty$. The quantum chaotic kicked top model for various Hilbert space dimensions is studied, and it is found to satisfy our conjecture. This involves the numerical verification of $m=2$ case of COE results. Our result can be used as a tool for the characterization of a system and to determine the symmetry structure of the system without desymmetrization of the spectra. Additionally, the comparative study of the higher-order spacing and ratio distributions in both $m=1$ and $m=2$ cases of COE as well as GOE is performed within and across these ensembles numerically using the $D(\beta')$ method. This study is carried out both by varying the dimension and keeping the number of realizations constant, and vice-versa. The same asymptotic higher-order statistics are observed across COE and GOE in terms of a given spectral fluctuation measure. But, within a given ensemble of COE or GOE, the results of higher-order spacing and ratio distributions agree with each other only up to some lower $k$, and beyond that, they start deviating from each other. Further, the spectral fluctuations of the intermediate map of various dimensions are studied. Various important observations and discussions from the analysis of our extensive numerical computations are presented.
Abstract:Higher-order spacing statistics in the $m$ superposed spectra of circular random matrices of the same class are studied numerically. We conjecture that for given $m$ (or order $k$) and $\beta$, the sequence of modified Dyson index $\beta'(k)$ (or $\beta'(m)$) obtained using the sum of absolute differences between the cumulative distribution functions method (denoted as $D(\beta')$) is unique. Also, for a given $k$, the distribution tends to the corresponding $k$-th order Poisson statistics in the limit $m\rightarrow \infty$. The quantum chaotic kicked top model for various Hilbert space dimensions is studied, and it is found to satisfy our conjecture. This involves the numerical verification of $m=2$ case of COE results. Our result can be used as a tool for the characterization of a system and to determine the symmetry structure of the system without desymmetrization of the spectra. Additionally, the comparative study of the higher-order spacing and ratio distributions in both $m=1$ and $m=2$ cases of COE as well as GOE is performed within and across these ensembles numerically using the $D(\beta')$ method. This study is carried out both by varying the dimension and keeping the number of realizations constant, and vice-versa. The same asymptotic higher-order statistics are observed across COE and GOE in terms of a given spectral fluctuation measure. But, within a given ensemble of COE or GOE, the results of higher-order spacing and ratio distributions agree with each other only up to some lower $k$, and beyond that, they start deviating from each other. Further, the spectral fluctuations of the intermediate map of various dimensions are studied. Various important observations and discussions from the analysis of our extensive numerical computations are presented.
24 pages (two-column) + 27 pages (one-column) + 77 figures. This is a substantially improved version of the previous manuscript, arXiv:2510.00503v1 [this http URL-an]. Comments are welcome
Data Analysis, Statistics and Probability (physics.data-an); Chaotic Dynamics (nlin.CD); Quantum Physics (quant-ph); Other Statistics (stat.OT)
arXiv:2510.00503 [physics.data-an]
(or arXiv:2510.00503v2 [physics.data-an] for this version)
From: Udaysinh T. Bhosale [view email] [v1] Wed, 1 Oct 2025 04:34:11 UTC (2,670 KB) [v2] Tue, 2 Jun 2026 09:59:22 UTC (1,469 KB)
Full text under the publisher's open licence · 348 words