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

Abstract

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 $N\to \infty$ is known to be a $\tau$-function of the fifth Painlevé 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.

23 pages, 2 figures, to appear in proceedings of Symmetries and Integrability of Difference Equations VII, Melbourne 2006

References in corpus (3)

• Random Matrix Theory and the Sixth Painlevé Equation
• An Isomonodromy Cluster of Two Regular Singularities
• The Widom-Dyson constant for the gap probability in random matrix theory