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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: [email protected].
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.
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Abstract

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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.

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

References

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