Level-Spacing Distributions and the Bessel Kernel
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Level-Spacing Distributions and the Bessel Kernel

  • Author(s): Tracy, Craig A.
  • Widom, Harold
  • et al.

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The level spacing distributions which arise when one rescales the Laguerre or Jacobi ensembles of hermitian matrices is studied. These distributions are expressible in terms of a Fredholm determinant of an integral operator whose kernel is expressible in terms of Bessel functions of order $\alpha$. We derive a system of partial differential equations associated with the logarithmic derivative of this Fredholm determinant when the underlying domain is a union of intervals. In the case of a single interval this Fredholm determinant is a Painleve tau function.

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