Boundary conditions for scaled random matrix ensembles in the bulk of the spectrum

A spectral average which generalizes the local spacing distribution of the eigenvalues of random N x N Hermitian matrices in the bulk of their spectrum as N → ∞ is known to be a τ-function of the fifth Painleve system. This τ-function, τ(s), has generic parameters and is transcendental but is characterized 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 the τ-function is of the separatrix, or partially truncated type