Power spectrum of the circular unitary ensemble

TL;DR

This paper derives a universal, explicit formula for the power spectrum of eigen-angles in the circular unitary ensemble, connecting it to Painlevé functions and random point fields, with implications for quantum chaos.

Contribution

It provides a novel, concise, parameter-free formula for the power spectrum of CUE eigen-angles involving Painlevé transcendent functions.

Findings

Derived a formula involving Painlevé functions for the power spectrum.

Established universality of the power spectrum law.

Tabulated the power spectrum for practical use.

Abstract

We study the power spectrum of eigen-angles of random matrices drawn from the circular unitary ensemble ${\rm CUE}(N)$ and show that it can be evaluated in terms of either a Fredholm determinant, or a Toeplitz determinant, or a sixth Painlev\'e function. In the limit of infinite-dimensional matrices, $N\rightarrow\infty$, we derive a ${\it\, concise\,}$ parameter-free formula for the power spectrum which involves a fifth Painlev\'e transcendent and interpret it in terms of the ${\rm Sine}_2$ 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.