Abstract
The eigenvalue probability density functions of the classical random matrix ensembles have a well known analogy with the one component log-gas at the special couplings β=1,2 and 4. It has been known for some time that there is an exactly solvable two-component log-potential plasma which interpolates between the β=1 and 4 circular ensemble, and an exactly solvable two-component generalized plasma which interpolates between β=2 and 4 circular ensemble. We extend known exact results relating to the latter—for the free energy and one and two-point correlations—by giving the general (k 1+k 2)-point correlation function in a Pfaffian form. Crucial to our working is an identity which expresses the Vandermonde determinant in terms of a Pfaffian. The exact evaluation of the general correlation is used to exhibit a perfect screening sum rule.
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References
Arikawa, M., Yamamoto, T., Saiga, Y., Kuramoto, Y.: Spin dynamics in the supersymmetric t–J model with inverse-square interaction. J. Phys. Soc. Jpn. 73, 808–811 (2004)
Borodin, A., Sinclair, C.D.: The Ginibre ensemble of real random matrices and its scaling limit. Commun. Math. Phys. 291, 177–224 (2009)
Dyson, F.J.: Statistical theory of energy levels of complex systems I. J. Math. Phys. 3, 140–156 (1962)
Dyson, F.J.: Correlations between the eigenvalues of a random matrix. Commun. Math. Phys. 19, 235–250 (1970)
Forrester, P.J.: An exactly solvable two-component classical Coulomb system. J. Aust. Math. Soc. Ser. B 26, 119–128 (1984)
Forrester, P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton (2010)
Forrester, P.J., Jancovici, B.: Generalized plasmas and the anomalous quantum hall effect. J. Phys. Lett. 45, L583–L589 (1984)
Forrester, P.J., Mays, A.: Pfaffian point process for the Gaussian real generalised eigenvalue problem, arXiv:0910.2531
Halperin, B.I.: Theory of the quantized Hall conductance. Helv. Phys. Acta 56, 75–102 (1983)
Ishikawa, M., Okanda, S., Tagawa, H., Zeng, J.: Generalizations of Cauchy’s determinant and Schur’s Pfaffian. Adv. Appl. Math. 36, 251–287 (2006)
Krivnov, V.Y., Ovchinnikov, A.A.: An exactly solvable one-dimensional problem with several particle species. Theor. Math. Phys. 50, 100–103 (1982)
Kuramoto, Y., Kato, Y.: Dynamics of One-Dimensional Quantum Systems: Inverse Square Interaction Models. Cambridge University Press, Cambridge (2009)
Mays, A.: PhD thesis, University of Melbourne, in preparation
Mehta, M.L.: A note on correlations between eigenvalues of random matrices. Commun. Math. Phys. 20, 245–250 (1971)
Mehta, M.L., Dyson, F.J.: Statistical theory of the energy levels of complex systems. V. J. Math. Phys. 4, 713–719 (1963)
Rider, B., Sinclair, C.D., Xu, Y.: A solvable mixed charge ensemble on the line: global results. arXiv:1007.2246 (2010)
Sinclair, C.D.: Ensemble averages when β is a square integer. arXiv:1008.4362 (2010)
Sutherland, B.: Quantum many-body problem in one dimension. J. Math. Phys. 12, 246–250 (1971)
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Forrester, P.J., Sinclair, C.D. A Generalized Plasma and Interpolation Between Classical Random Matrix Ensembles. J Stat Phys 143, 326–345 (2011). https://doi.org/10.1007/s10955-011-0173-3
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DOI: https://doi.org/10.1007/s10955-011-0173-3