Random matrix theory has applications
in a number of diverse fields. For example, in analytic number
theory the Montgomery-Odlyzko law states that the statistics of
the large zeros of the Riemann zeta function on the critical line
coincide with the statistics of the eigenvalues of large random
unitary matrices. In combinatorics the distribution of the longest
increasing subsequence of a large random permutation coincides
with the distribution of the largest eigenvalue of a large random
complex Hermitian matrix. Quantitative results for eigenvalue
distributions, and thus for the distributions in the applications,
can be obtained in terms of Painlevé transcendents. We
will emphasize the approach to such calculations via a Hamiltonian
formulation of Painlevé systems.
