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arXiv · 1304.4368

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

Yan V Fyodorov·Boris A Khoruzhenko·André Nock
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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 ${\cal P}(K)\propto \left[\det{(λ^2+(K-ε)^2)}\right]^{-M}$ 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.

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Yan V Fyodorov, Boris A Khoruzhenko, André Nock. 2013-07-19. Universal K-matrix distribution in beta=2 Ensembles of Random Matrices. https://doi.org/10.1088/1751-8113/46/26/262001

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