ANZIAM J.
44 (2002), 41-50
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Integrability, random matrices and Painlevé transcendents
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N. S. Witte
Department of Mathematics
and Statistics
and School of Physics
University of Melbourne
VIC 3010
Australia
nsw@ms.unimelb.edu.au
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P. J. Forrester
Department of Mathematics
and Statistics
University of Melbourne
VIC 3010
Australia
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C. M. Cosgrove
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.
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