*(English)*Zbl 1042.82019

Summary: *C. A. Tracy* and *H. Widom* [Commun. Math. Phys. 159, 151–174 (1994; Zbl 0789.35152))] have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite GUE in terms of a PTV and PII transcendent respectively. We generalise these results to the evaluation of

where ${\chi}_{(-\infty ,\lambda ]}^{\left(l\right)}=1$ for ${\lambda}_{l}\in (-\infty ,\lambda ]$ and ${\chi}_{(-\infty ,\lambda ]}^{\left(l\right)}=0$ otherwise, and the average is with respect to the joint eigenvalue distribution of the GUE, as well as to the evaluation of

Of particular interest are ${\tilde{E}}_{N}(\lambda ;2)$ and ${F}_{N}(\lambda ;2)$, and their scaled limits, which gives the distribution of the largest eigenvalue and the density respectively. Our results are obtained by applying the Okamoto $\tau $-function theory of PIV and PII [cf, *K. Okamoto*, [Studies on the Painlevé Equations. III. Second and Fourth Painlevé Equations, ${P}_{II}$ and ${P}_{IV}$. Math. Ann. 275, 221–255 (1986; Zbl 0589.58008)], for which we give a self contained presentation based on the recent work of *M. Noumi* and *Y. Yamada* [Nagoya Math. J. 153, 53–86 (1999; Zbl 0932.34088)]. We point out that the same approach can be used to study the quantities ${\tilde{E}}_{N}(\lambda ;a)$ and ${F}_{N}(\lambda ;a)$ for the other classical matrix ensembles.