Level compressibility of certain random unitary matrices

TL;DR

This paper analytically calculates the level compressibility of certain random unitary matrices, revealing it equals 1/2 for matrices from barrier billiards, indicating intermediate spectral statistics.

Contribution

It provides a rigorous proof that the level compressibility for matrices from barrier billiards is exactly 1/2, regardless of barrier parameters.

Findings

Level compressibility of barrier billiard matrices is 1/2.

Method based on eigenvalues of transition matrices derived from matrix element moduli.

Analytical approach applicable to models with intermediate spectral statistics.

Abstract

The value of spectral form factor at the origin, called level compressibility, is an important characteristic of random spectra. The paper is devoted to analytical calculations of this quantity for different random unitary matrices describing models with intermediate spectral statistics. The computations are based on the approach developed by G. Tanner in [J. Phys. A: Math. Gen. 34, 8485 (2001)] for chaotic systems. The main ingredient of the method is the determination of eigenvalues of a transition matrix whose matrix elements equal squared moduli of matrix elements of the initial unitary matrix. The principal result of the paper is the proof that the level compressibility of random unitary matrices derived from the exact quantisation of barrier billiards and consequently of barrier billiards themselves is equal to $1/2$ irrespectively of the height and the position of the barrier.