Gap probabilities in non-Hermitian random matrix theory

摘要

We compute the gap probability that a circle of radius r around the origin containsexactly k complex eigenvalues. Four different ensembles of random matrices areconsidered: the Ginibre ensembles and their chiral complex counterparts, with bothcomplex (β=2) or quaternion real (β=4) matrix elements. For general non-Gaussian weights we give a Fredholm determinant or Pfaffian representation re-spectively, depending on the non-Hermiticity parameter. At maximal non-Hermiticity, that is, for rotationally invariant weights, the product of Fredholmeigenvalues 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 choos-ing Gaussian weights we give new explicit expressions for the Fredholm eigenval-ues in the chiral case, in terms of Bessel-K and incomplete Bessel-I functions. Thiscompares with known results for the Ginibre ensembles in terms of incompleteexponentials. Furthermore, we present an asymptotic expansion of the logarithm ofthe gap probability for large argument r at large N in all four ensembles, up to andincluding the third order linear term. We can provide strict upper and lower boundsand present numerical evidence for the conjectured values of the linear term, de-pending on the number of exact zero eigenvalues in the chiral ensembles. For theGinibre ensemble at β=2, exact results were previously derived by Forrester [Phys.Lett. A 169, 21 (1992)].