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

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

2007-06-04

arxiv.org/abs/0706.0359v1

Mathematics

Classical Analysis and ODEs

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

Scientific paper

10.1088/1751-8113/40/42/S16

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 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.

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