Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-15T02:02:13.075Z Has data issue: false hasContentIssue false
  • English
  • Français

Integrability, random matrices and Painlevé transcendents

Published online by Cambridge University Press:  17 February 2009

N. S. Witte,
P. J. Forrester and
Christopher M. Cosgrove
N. S. Witte
Affiliation:
Department of Mathematics and Statistics and School of Physics, University of Melbourne, VIC 3010, Australia; e-mail: nsw@ms.unimelb.edu.au.
P. J. Forrester
Affiliation:
Department of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia.
Christopher M. Cosgrove
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney NSW 2006, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

The probability that an interval I is free of eigenvalues in a matrix ensemble with unitary symmetry is given by a Fredholm determinant. When the weight function in the matrix ensemble is a classical weight function, and the interval I includes an endpoint of the support, Tracy and Widom have given a formalism which gives coupled differential equations for the required probability and some auxiliary quantities. We summarize and extend earlier work by expressing the probability and some of the auxiliary quantities in terms of Painlevé transcendents.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 1 , July 2002, pp. 41 - 50
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Cosgrove, C. M. and Scoufis, G., “Painlevé classification of a class of differential equations of the second order and second degree”, Stud. Appl. Math. 88 (1993) 25–87.CrossRefGoogle Scholar
[2]Guhr, T., Müller-Groeling, A. and Weidenmüller, H. A., “Random-matrix theories in quantum physics: common concepts”, Phys. Rep. 299 (1999) 189–425.CrossRefGoogle Scholar
[3]Haine, L. and Semengue, J.-P., “The Jacobi polynomial ensemble and the Painlevé VI equation”, J. Math. Phys. 40 (1999) 2117–2134.CrossRefGoogle Scholar
[4]Mehta, M. L., Random matrices, 2nd ed. (Academic Press, San Diego, 1991).Google Scholar
[5]Montgomery, H. L., “The pair correlation of zeros of the zeta function”, in Proc. Sympos. Pure Math. vol. 24, (Amer. Math. Soc., Providence, RI, 1973) 181–193.Google Scholar
[6]Odlyzko, A. M., “The 1020th zero of the Riemann zeta function and 70 million of its neighbours”, preprint, AT&T Bell Laboratories, Murray Hill, NJ 07974, 1989.Google Scholar
[7]Tracy, C. A. and Widom, H., “Fredholm determinants, differential equations and matrix models”, Commun. Math. Phys. 163 (1994) 33–72.CrossRefGoogle Scholar
[8]Whittaker, E. T. and Watson, G. N., A course of modern analysis, 2nd ed. (Cambridge University Press, Cambridge, 1965).Google Scholar
[9]Witte, N. S. and Forrester, P. J., “Gap probabilities in the finite and scaled Cauchy random matrix ensembles”, Nonl. 13 (2000) 1965–1986.Google Scholar