Random matrices associated with general barrier billiards

TL;DR

This paper derives random unitary matrices that replicate the spectral statistics of quantum eigenvalues in certain two-dimensional barrier billiards, revealing universal semi-Poisson statistics for their local spectral correlations.

Contribution

The paper introduces a novel method to derive random matrices from barrier billiards using a high-energy phase approximation and Wiener-Hopf method, connecting deterministic billiard spectra with random matrix theory.

Findings

Spectral statistics of the billiards match semi-Poisson distribution.

Derived random matrices are similar to previous models but with different parameters.

Spectral correlations are universal across parameter ranges.

Abstract

The paper is devoted to the derivation of random unitary matrices whose spectral statistics is the same as statistics of quantum eigenvalues of certain deterministic two-dimensional barrier billiards. These random matrices are extracted from the exact billiard quantisation condition by applying a random phase approximation for high-excited states. An important ingredient of the method is the calculation of $S$-matrix for the scattering in the slab with a half-plane inside by the Wiener-Hopf method. It appears that these random matrices have the form similar to the one obtained by the author in [arXiv:2107.03364] for a particular case of symmetric barrier billiards but with different choices of parameters. The local correlation functions of the resulting random matrices are well approximated by the semi-Poisson distribution which is a characteristic feature of various models with intermediate statistics. Consequently, local spectral statistics of the considered barrier billiards is (i) universal for almost all values of parameters and (ii) well described by the semi-Poisson statistics.