Skew-Orthogonal Polynomials and Universality of Energy-Level Correlations

Abstract

Random-matrix ensembles serve as models for quantum chaotic systems. We develop the theory of skew-orthogonal polynomials to study matrix ensembles with non-Gaussian weight functions. From the asymptotic properties of these and the orthogonal polynomials, we show that the local energy level correlations in the ensembles become universal properties independent of the global level density. This provides a rigorous justification for the universality of the Gaussian ensemble results observed in quantum chaotic systems.

Random Matrix Model for Nakagami–Hoyt Fading

“…For τ = 0 (q = 0) and τ = ∞ (q = 1) equation (10) reduces to the following well known results for the LOE and LUE respectively [36], [37], [38], [39], [40], [41], [42],…”

“…The antisymmetric function G (0) (x, y) = 1 2 sgn(x − y) can be expanded in terms of the functions dual to the weighted skew-orthogonal polynomials [40], [41], [44], as defined in Appendix C. We have…”

“…The choice of probability distribution for H depends on the propagation environment and communication scenario. When the matrix elements of H are independent and identically distributed (iid) Gaussian variables with zero mean, the ensemble of W is referred to as the Wishart or equivalently Laguerre ensemble [36], [37], [38], [39], [40], [41]. It has also been referred to as the Dyson ensemble [42], [43].…”

“…In the context of random matrix theory Rayleigh fading corresponds to the Laguerre unitary ensemble (LUE) [38], [39], [40], [41] of random matrices, whereas one sided Gaussian fading corresponds to the Laguerre orthogonal ensemble (LOE) of random matrices [38], [39], [40], [41]. Thus Nakagami-q fading, in general, corresponds to a random matrix ensemble which interpolates between LOE and LUE.…”

Emerging spectra of singular correlation matrices under small power-map deformations

“…When expressed in terms of unit average spacing the npoint correlation functions, for large matrices, are the same as those for Gaussian orthogonal ensemble (GOE) [41], and therefore WOE spectra have universal spectral fluctuations [46]. In the case of actual correlations among the matrix elements A jk , one defines CWOE [1].…”