Universal K-matrix distribution in beta=2 Ensembles of Random Matrices

TL;DR

This paper proves that for beta=2 invariant ensembles of large random Hermitian matrices, the K-matrix distribution converges to a matrix Cauchy distribution, confirming a universality conjecture in quantum chaos.

Contribution

The authors explicitly demonstrate the universality of the K-matrix distribution for beta=2 ensembles, extending understanding of spectral properties in quantum chaotic systems.

Findings

K-matrix distribution converges to a matrix Cauchy distribution for beta=2 ensembles.

Universality holds for a broad class of invariant ensembles as N approaches infinity.

Finite blocks of the resolvent are Cauchy distributed in these ensembles.

Abstract

The K-matrix, also known as the "Wigner reaction matrix" in nuclear scattering or "impedance matrix" in the electromagnetic wave scattering, is given essentially by an M x M diagonal block of the resolvent (E-H)^{-1} of a Hamiltonian H. For chaotic quantum systems the Hamiltonian H can be modelled by random Hermitian N x N matrices taken from invariant ensembles with the Dyson symmetry index beta=1,2,4. For beta=2 we prove by explicit calculation a universality conjecture by P. Brouwer which is equivalent to the claim that the probability distribution of K, for a broad class of invariant ensembles of random Hermitian matrices H, converges to a matrix Cauchy distribution with density ${\cal P}(K)\propto \left[\det{({\lambda}^2+(K-{\epsilon})^2)}\right]^{-M}$ in the limit $N\to \infty$, provided the parameter M is fixed and the spectral parameter E is taken within the support of the eigenvalue distribution of H. In particular, we show that for a broad class of unitary invariant ensembles of random matrices finite diagonal blocks of the resolvent are Cauchy distributed. The cases beta=1 and beta=4 remain outstanding.