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2009

Gap Probabilities in Non-Hermitian Random Matrix Theory

Akemann, G., Phillips, M. J., Shifrin, L.

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We compute the gap probability that a circle of radius r around the origin contains exactly k complex eigenvalues.

Four different ensembles of random matrices are considered: the Ginibre ensembles and their chiral complex counterparts, with both complex (beta=2) or quaternion real (beta=4) matrix elements.
For general non-Gaussian weights we give a Fredholm determinant or Pfaffian representation respectively, depending on the non-Hermiticity parameter.
At maximal non-Hermiticity, that is for rotationally invariant weights, the product of Fredholm eigenvalues for beta=4 follows from beta=2 by skipping every second factor, in contrast to the known relation for Hermitian ensembles.

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