Higher-order level spacings in random matrix theory based on Wigner's conjecture

Abstract

The distribution of higher order level spacings, i.e. the distribution of $\{s_i^{(n)}=E_{i+n}-E_i\}$ with $n\ge 1$ is derived analytically using a Wigner-like surmise for Gaussian ensembles of random matrix as well as Poisson ensemble. It is found $s^{(n)}$ in Gaussian ensembles follows a generalized Wigner-Dyson distribution with rescaled parameter $\alpha =\nu C_{n+1}^2+n-1$, while that in Poisson ensemble follows a generalized semi-Poisson distribution with index $n$. Numerical evidences are provided through simulations of random spin systems as well as non-trivial zeros of Riemann zeta function. The higher order generalizations of gap ratios are also discussed.

arXiv comment7 pages, 4 figures

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• Random Matrix Theory of the Isospectral twirling
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• Decomposing the Spectral Form Factor
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• Higher-order gap ratios of singular values in open quantum systems
• Random Matrix Model for Eigenvalue Statistics in Random Spin Systems
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• Comparative Study of Indicators of Chaos in the Closed and Open Dicke Model