## Data Repository

• Power Spectrum in Random Matrix Theory and Quantum Chaology

• ☘  Infinite dimensional circular unitary ensemble ($$\beta = 2$$)
•

Data file.  The two-column file  cue_data.csv [25 KB]     tabulates the power spectrum  $$S_\infty (\omega)$$  specified in Theorem 1.2 of the paper  RK-2022  .  The first column contains  $$999$$  rescaled frequencies  $$\tilde\omega = \omega/2\pi$$  in the interval from  $$5\times 10^{-4}$$  to   $$0.5 - 5\times 10^{-4}$$   with the increment  $$5\times 10^{-4}$$. The second column contains the power spectrum computed numerically to an absolute and relative errors better than  $$\delta_{\rm abs} = 10^{-6}$$  and  $$\delta_{\rm rel} = 10^{−5}$$, respectively.

Technical remarks.  For frequencies away from the edges  $$\tilde\omega = 0$$  and  $$\tilde\omega_{\rm Ny} = 1/2$$,  the power spectrum was computed with an ODE solver for the fifth Painlevé transcendent.   At very small frequencies, the data were produced from the small-$$\omega$$  expansion proved in Proposition 4.10.   Close to the Nyquest frequency  $$\tilde\omega_{\rm Ny} = 1/2$$,  a numerical solution to the fifth Painlevé transcendent becomes unstable due to the poles of  $$\sigma_0 ( \lambda;\zeta )$$   in the complex $$\lambda$$-plane close to the real axis. There, an evaluation of the power spectrum was based on the Bornemann code  [F.  Bornemann,  «On the numerical evaluation of Fredholm determinants».  Math. Comp.  79,  871  (2010)].