Crossover ensembles of random matrices and skew-orthogonal polynomials
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110 067 (India)
Highlights: > We study crossover ensembles of Jacobi family of random matrices. > We consider correlations for orthogonal-unitary and symplectic-unitary crossovers. > We use the method of skew-orthogonal polynomials and quaternion determinants. > We prove universality of spectral correlations in crossover ensembles. > We discuss applications to quantum conductance and communication theory problems. - Abstract: In a recent paper (S. Kumar, A. Pandey, Phys. Rev. E, 79, 2009, p. 026211) we considered Jacobi family (including Laguerre and Gaussian cases) of random matrix ensembles and reported exact solutions of crossover problems involving time-reversal symmetry breaking. In the present paper we give details of the work. We start with Dyson's Brownian motion description of random matrix ensembles and obtain universal hierarchic relations among the unfolded correlation functions. For arbitrary dimensions we derive the joint probability density (jpd) of eigenvalues for all transitions leading to unitary ensembles as equilibrium ensembles. We focus on the orthogonal-unitary and symplectic-unitary crossovers and give generic expressions for jpd of eigenvalues, two-point kernels and n-level correlation functions. This involves generalization of the theory of skew-orthogonal polynomials to crossover ensembles. We also consider crossovers in the circular ensembles to show the generality of our method. In the large dimensionality limit, correlations in spectra with arbitrary initial density are shown to be universal when expressed in terms of a rescaled symmetry breaking parameter. Applications of our crossover results to communication theory and quantum conductance problems are also briefly discussed.
- OSTI ID:
- 21583318
- Journal Information:
- Annals of Physics (New York), Vol. 326, Issue 8; Other Information: DOI: 10.1016/j.aop.2011.04.013; PII: S0003-4916(11)00072-8; Copyright (c) 2011 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; ISSN 0003-4916
- Country of Publication:
- United States
- Language:
- English
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