All the lowest order PDE for spectral gaps of Gaussian matrices

Details All the lowest order PDE for spectral gaps of Gaussian matrices All the lowest order PDE for spectral gaps of Gaussian matrices

2010-08-20

arxiv.org/abs/1008.3560v2

Physics

Mathematical Physics

22 pages, references corrected, remark added

Scientific paper

Tracy-Widom (TW) equations for one-matrix unitary ensembles (UE) (equivalent to a particular case of Schlesinger equations for isomonodromic deformations) are rewritten in a general form which allows one to derive all the lowest order equations (PDE) for spectral gap probabilities of UE without intermediate higher-order PDE. This is demonstrated on the example of Gaussian ensemble (GUE) for which all the third order PDE for gap probabilities are obtained explicitly. Moreover, there is a {\it second order} PDE for GUE probabilities in the case of more than one spectral endpoint. This approach allows to derive all PDE at once where possible, while in the method based on Hirota bilinear identities and Virasoro constraints starting with different bilinear identities leads to different subsets of the full set of equations.

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