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

P. J. Forrester, N. S. Witte

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,μ) := < \prod_{l=1}^N χ_{(0,\i…