Application of the $τ$-function theory of Painlevé equations to random matrices: \PV, \PIII, the LUE, JUE and CUE

With $<\cdot>$ denoting an average with respect to the eigenvalue PDF for the Laguerre unitary ensemble, the object of our study is $ \tilde{E}_N(I;a,\mu) := < \prod_{l=1}^N \chi_{(0,\infty)\backslash I}^{(l)} (\lambda - \lambda_l)^\mu>$ for $I = (0,s)$ and $I = (s,\infty)$, where $\chi_I^{(l)} = 1$ for $ \lambda_l \in I$ and $\chi_I^{(l)} = 0$ otherwise. Using Okamoto's development of the theory of the Painlev\'e V equation, it is shown that $\tilde{E}_N(I;a,\mu)$ is a $\tau$-function associated with the Hamiltonian therein, and so can be characterised as the solution of a certain second order second degree differential equation, or in terms of the solution of certain difference equations. The cases $\mu = 0$ and $\mu = 2$ are of particular interest as they correspond to the cumulative distribution and density function respectively for the smallest and largest eigenvalue. In the case $I = (s,\infty)$, $\tilde{E}_N(I;a,\mu)$ is simply related to an average in the Jacobi unitary ensemble, and this in turn is simply related to certain averages over the orthogonal group, the unitary symplectic group and the circular unitary ensemble. The latter integrals are of interest for their combinatorial content. Also considered are the hard edge and soft edge scaled limits of $\tilde{E}_N(I;a,\mu)$. In particular, in the hard edge scaled limit it is shown that the limiting quantity $E^{\rm hard}((0,s);a,\mu)$ can be evaluated as a $\tau$-function associated with the Hamiltonian in Okamoto's theory of the Painlev\'e III equation.