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Application of the -Function Theory of Painlev\'e Equations to Random Matrices: PIV, PII and the GUE
Tracy and Widom have evaluated the cumulative distribution of the largest
eigenvalue for the finite and scaled infinite GUE in terms of a PIV and PII
transcendent respectively. We generalise these results to the evaluation of
, where
for and otherwise, and the average is with respect to the joint eigenvalue
distribution of the GUE, as well as to the evaluation of F_N(\lambda;a) :=
\Big . Of particular interest
are and , and their scaled limits,
which give the distribution of the largest eigenvalue and the density
respectively. Our results are obtained by applying the Okamoto -function
theory of PIV and PII, for which we give a self contained presentation based on
the recent work of Noumi and Yamada. We point out that the same approach can be
used to study the quantities and for
the other classical matrix ensembles.Comment: 40 pages, Latex2e plus AMS and XY packages. to appear Commun. Math.
Phy
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