Boundary Conditions for Scaled Random Matrix Ensembles in the Bulk of the Spectrum

A spectral average which generalises the local spacing distribution of the eigenvalues of random $N\times N$ hermitian matrices in the bulk of their spectrum as $N\to \mathrm{\infty }$ is known to be a $\tau$-function of the fifth Painlev'e system. This $\tau$-function, $\tau (s)$, has generic parameters and is transcendental but is characterised by particular boundary conditions about the singular point $s=0$, which we determine here. When the average reduces to the local spacing distribution we find that $\tau$-function is of the separatrix, or partially truncated type.