## Abstract

We consider eigenvalues of a product of n non-Hermitian, independent random matrices. Each matrix in this product is of size N×N with independent standard complex Gaussian variables. The eigenvalues of such a product form a determinantal point process on the complex plane (Akemann and Burda in J. Phys. A, Math. Theor. 45:465201, 2011), which can be understood as a generalization of the finite Ginibre ensemble. As N→∞, a generalized infinite Ginibre ensemble arises. We show that the set of absolute values of the points of this determinantal process has the same distribution as {R_{1}^{(n)}, R_{2}^{(n)}, ...}, where R_{k}^{(n)} are independent, and (R^{(n)}_{k}})^{2} is distributed as the product of n independent Gamma variables Gamma (k, 1). This enables us to find the asymptotics for the hole probabilities, i. e. for the probabilities of the events that there are no points of the process in a disc of radius r with its center at 0, as r → ∞. In addition, we solve the relevant overcrowding problem: we derive an asymptotic formula for the probability that there are more than m points of the process in a fixed disk of radius r with its center at 0, as m → ∞.

Original language | American English |
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Pages (from-to) | 987-1003 |

Number of pages | 17 |

Journal | Journal of Statistical Physics |

Volume | 151 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2013 |

### Bibliographical note

Funding Information:The first author (G.A.) is partly supported by the SFB|TR12 “Symmetries and Universality in Mesoscopic Systems” of the German research council DFG. The second author (E.S.) is supported in part by the US-Israel Binational Science Foundation (BSF) Grant No. 2006333, and by the Israel Science Foundation (ISF) Grant No. 1441/08.

## Keywords

- Determinantal processes
- Generalized Ginibre ensembles
- Hole probabilities
- Non-Hermitian random matrix theory
- Overcrowding
- Products of random matrices