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

##### View/Open

##### Volume

46

##### Pagination

26001 - 26011 (10)

##### DOI

10.1088/1751-8113/46/26/262001

##### Issue

##### ISSN

1751-8113

##### Metadata

Show full item record##### 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.

##### Authors

Fyodorov, YV; Khoruzhenko, BA; Nock, A##### Collections

- Applied Mathematics [115]