On the bilinear equations for Fredholm determinants appearing in random matrices

Details On the bilinear equations for Fredholm determinants appearing in random matrices On the bilinear equations for Fredholm determinants appearing in random matrices

1999-06-10

arxiv.org/abs/solv-int/9906004v3

J. Nonlinear Math. Phys., volume 9, no. 4 (2002) 530-550

Physics

High Energy Physics

High Energy Physics - Theory

arxiv version is already official

Scientific paper

It is shown how the bilinear differential equations satisfied by Fredholm determinants of integral operators appearing as spectral distribution functions for random matrices may be deduced from the associated systems of nonautonomous Hamiltonian equations satisfied by auxiliary canonical phase space variables introduced by Tracy and Widom. The essential step is to recast the latter as isomonodromic deformation equations for families of rational covariant derivative operators on the Riemann sphere and interpret the Fredholm determinants as isomonodromic $\tau$-functions.

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