Abstract
Correlations for parameter-dependent Gaussian random matrices, intermediate between symmetric and Hermitian and antisymmetric Hermitian and Hermitian, are calculated. The (dynamical) density-density correlation between eigenvalues at different values of the parameter is calculated for the symmetric to Hermitian transition and the scaledN→∞ limit is computed. For the antisymmetric Hermitian to Hermitian transition the equal-parametern-point distribution function is calculated and the scaled limit computed. A circular version of the antisymmetric Hermitian to Hermitian transition is formulated. In the thermodynamic limit the equal-parameter distribution function is shown to coincide with the scaled-limit expression of this distribution for the Gaussian antisymmetric Hermitian to Hermitian transition. Furthermore, the thermodynamic limit of the corresponding density-density correlation is computed. The results for the correlations are illustrated by comparison with empirical correlations calculated from numerical data obtained from computer-generated Gaussian random matrices.
Similar content being viewed by others
References
A. Pandey and M. L. Mehta,Commun. Math. Phys. 87:449 (1983); M. L. Mehta and A. Pandey,J. Phys. A 16 :2655 (1983).
G. Lenz and K. Zyczkowski,J. Phys. A 25:5539 (1992).
F. J. Dyson,J. Math. Phys. 3:1191 (1962).
M. L. Mehta,Commun. Math. Phys. 20:245 (1971).
P. J. Forrester,Physica A 223:365 (1996).
K. Frahm and J. L. Pichard,J. Phys. I (France) 5:877 (1995).
M. L. Mehta,Random Matrices 2nd ed., (Academic, New York, 1991).
F. Haake,Quantum Signatures of Chaos, (Springer, Berlin, 1992).
C. W. J. Beenakker and B. Rejaei,Physica A 203:61 (1994).
T. Nagao and M. Wadati,J. Phys. Soc. Japan 60:298 (1991).
T. Nagao and P. J. Forrester,Nucl. Phys B 435:401 (1995).
F. J. Dyson,Commun. Math. Phys. 19:235 (1970).
A. Pandey and P. Shukla,J. Phys. A 24:907 (1991).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Forrester, P.J., Nagao, T. Correlations for parameter— dependent random matrices. J Stat Phys 89, 69–110 (1997). https://doi.org/10.1007/BF02770755
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02770755