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

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 $\mathcal{P}(K)\propto {\left[\det ({\lambda }^2+(K-\epsilon {)}^2)\right]}^{-M}$ 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.

11 pages; published version (added proportionality constants, minor changes)

References in corpus (1)

• Scattering, reflection and impedance of waves in chaotic and disordered systems with absorption

Cited by in corpus (7)

• Anderson transition on the Bethe lattice: an approach with real energies
• Number statistics for $\beta$-ensembles of random matrices: applications to trapped fermions at zero temperature
• Distributions of Off-Diagonal Scattering Matrix Elements: Exact Results
• Index Distribution of Cauchy Random Matrices
• On Random Matrix Averages Involving Half-Integer Powers of GOE Characteristic Polynomials
• Statistics of quantum transport in weakly non-ideal chaotic cavities
• Asymptotic Coincidence of the Statistics for Degenerate and Non-Degenerate Correlated Real Wishart Ensembles