Hamiltonian Structure of Equations Appearing in Random Matrices
Open Access Publications from the University of California

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https://arxiv.org/pdf/hep-th/9301051.pdf
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Abstract

The level spacing distributions in the Gaussian Unitary Ensemble, both in the bulk of the spectrum,'' given by the Fredholm determinant of the operator with the sine kernel ${\sin \pi(x-y) \over \pi(x-y)}$ and on the edge of the spectrum,'' given by the Airy kernel ${\rm{Ai}(x) \rm{Ai}'(y) - \rm{Ai}(y) \rm{Ai}'(x) \over (x-y)}$, are determined by compatible systems of nonautonomous Hamiltonian equations. These may be viewed as special cases of isomonodromic deformation equations for first order $2\times 2$ matrix differential operators with regular singularities at finite points and irregular ones of Riemann index 1 or 2 at $\infty$. Their Hamiltonian structure is explained within the classical R-matrix framework as the equations induced by spectral invariants on the loop algebra ${\tilde{sl}(2)}$, restricted to a Poisson subspace of its dual space ${\tilde{sl}^*_R(2)}$, consisting of elements that are rational in the loop parameter.

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