UniversalK-matrix distribution in β = 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 × 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 × N matrices taken from invariant ensembles with the Dyson symmetry index β = 1, 2 or 4. For β = 2 we prove by explicit calculation a universality conjecture by P. Brouwer (Phys. Rev. B 51 (1995), 16878-84) 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 P(K) ∝ det (λ 2 + (K − ǫ) 2 ) −M in the limit N → ∞, 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 β = 1 and β = 4 remain outstanding.

Distributions of off-diagonal scattering matrix elements: Exact results

“…Since the universal properties depend only on the invariance properties of the underlying Hamiltonian [64][65][66], his result established the equivalence between the two approaches. Furthermore, very recently Fyodorov et al have demonstrated this equivalence for a broad class of unitary-invariant ensembles of random matrices [68].…”

On Random Matrix Averages Involving Half-Integer Powers of GOE Characteristic Polynomials

“…A similar formula for invariant ensembles of complex Hermitian random matrices H ( i.e. β = 2) was proved rigorously very recently in [18], and in the same paper it was mentioned that for β = 1 and the case of random Gaussian coupling the following relation holds §:…”

“…• As it has been mentioned above, we were not yet able to reveal nice mathematical structures for (1) at finite values of the matrix size N beyond the simplest case K = 1, L = 1, where the methods outlined below yielded a determinantal structure § The corresponding formula in [18] was written not accurately enough and did not show the dependence on sgn det factors.…”

“…Similar averages arise if one is interested in statistics of the matrix elements of the resolvents computed in the basis of random Gaussian vectors, as it is frequently done in applications to scattering systems with Quantum Chaos, see e.g. the recent paper [18] for an example and further references. For those and other reasons averages of products and ratios of powers of characteristic polynomials of random matrices attracted much interest over the years.…”

Asymptotic coincidence of the statistics for degenerate and non-degenerate correlated real Wishart ensembles

“…For instance, gap probabilities related to the smallest and largest eigenvalue were found to possess a representation as averaged products of determinants in the denominator to half integer power [17]. Other examples are the eigenvalue density in the ordinary and doubly correlated Wishart model [18][19][20][21], the distribution of the smallest eigenvalue [22][23][24][25] as well as universality considerations in scattering theory [26,27]. Those square roots are serious obstacles in analytical calculations and a solution is urgently called for.…”