Gap probabilities in non-Hermitian random matrix theory

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 ( β = 2 ) or quaternion real ( β = 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 β = 4 follows from the β = 2 case by skipping every second factor, in contrast to the known relation for Hermitian ensembles. On additionally choosing Gaussian weights we give new explicit expressions for the Fredholm eigenvalues in the chiral case, in terms of Bessel- K and incomplete Bessel- I functions. This compares with known results for the Ginibre ensembles in terms of incomplete exponentials. Furthermore, we present an asymptotic expansion of the logarithm of the gap probability for large argument r at large N in all four ensembles, up to and including the third order linear term. We can provide strict upper and lower bounds and present numerical evidence for the conjectured values of the linear term, depending on the number of exact zero eigenvalues in the chiral ensembles. For the Ginibre ensemble at β = 2 , exact results were previously derived by Forrester [Phys. Lett. A 169, 21 (1992)].