Power spectrum of the circular unitary ensemble

Abstract

We study the power spectrum of eigen-angles of random matrices drawn from the circular unitary ensemble $\mathrm{CUE}(N)$ and show that it can be evaluated in terms of either a Fredholm determinant, or a Toeplitz determinant, or a sixth Painlevé function. In the limit of infinite-dimensional matrices, $N\to \infty$, we derive a $concise$ parameter-free formula for the power spectrum which involves a fifth Painlevé transcendent and interpret it in terms of the ${\mathrm{Sine}}_2$ determinantal random point field. Further, we discuss a universality of the predicted power spectrum law and tabulate it (follow http://eugenekanzieper.faculty.hit.ac.il/data.html) for easy use by random-matrix-theory and quantum chaos practitioners.

47 pages; 4 figures; published version

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• Spectral fluctuations and 1/f noise in the order-chaos transition regime
• Sub-diffusive Thouless time scaling in the Anderson model on random regular graphs
• Nonperturbative theory of power spectrum in complex systems
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Cited by in corpus (5)

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• The Rosenzweig Porter model revisited for the three Wigner Dyson symmetry classes
• Power spectra and autocovariances of level spacings beyond the Dyson conjecture
• Statistics of local level spacings in single- and many-body quantum chaos
• The Tracy-Widom distribution at large Dyson index