# Gap Probabilities in Non-Hermitian Random Matrix Theory

Akemann G, Phillips MJ, Shifrin L (2009)

J. Math. Phys. 50(6).

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We compute the gap probability that a circle of radius r around the origincontains exactly k complex eigenvalues. Four different ensembles of randommatrices are considered: the Ginibre ensembles and their chiral complexcounterparts, with both complex (beta=2) or quaternion real (beta=4) matrixelements. For general non-Gaussian weights we give a Fredholm determinant orPfaffian representation respectively, depending on the non-Hermiticityparameter. At maximal non-Hermiticity, that is for rotationally invariantweights, the product of Fredholm eigenvalues for beta=4 follows from beta=2 byskipping every second factor, in contrast to the known relation for Hermitianensembles. On additionally choosing Gaussian weights we give new explicitexpressions for the Fredholm eigenvalues in the chiral case, in terms ofBessel-K and incomplete Bessel-I functions. This compares to known results forthe Ginibre ensembles in terms of incomplete exponentials. Furthermore wepresent an asymptotic expansion of the logarithm of the gap probability forlarge argument r at large N in all four ensembles, up to including the thirdorder linear term. We can provide strict upper and lower bounds and presentnumerical evidence for its conjectured values, depending on the number of exactzero eigenvalues in the chiral ensembles. For the Ginibre ensemble at beta=2exact results were previously derived by Forrester.

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Akemann G, Phillips MJ, Shifrin L. Gap Probabilities in Non-Hermitian Random Matrix Theory.

*J. Math. Phys.*2009;50(6).Akemann, G., Phillips, M. J., & Shifrin, L. (2009). Gap Probabilities in Non-Hermitian Random Matrix Theory.

*J. Math. Phys.*,*50*(6).Akemann, G., Phillips, M. J., and Shifrin, L. (2009). Gap Probabilities in Non-Hermitian Random Matrix Theory.

*J. Math. Phys.*50.Akemann, G., Phillips, M.J., & Shifrin, L., 2009. Gap Probabilities in Non-Hermitian Random Matrix Theory.

*J. Math. Phys.*, 50(6).G. Akemann, M.J. Phillips, and L. Shifrin, “Gap Probabilities in Non-Hermitian Random Matrix Theory”,

*J. Math. Phys.*, vol. 50, 2009.Akemann, G., Phillips, M.J., Shifrin, L.: Gap Probabilities in Non-Hermitian Random Matrix Theory. J. Math. Phys. 50, (2009).

Akemann, Gernot, Phillips, M. J., and Shifrin, L. “Gap Probabilities in Non-Hermitian Random Matrix Theory”.

*J. Math. Phys.*50.6 (2009).
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