## Abstract

We apply the recently introduced method of hermitization to study in the large N limit non-hermitian random matrices that are drawn from a large class of circularly symmetric non-gaussian probability distributions, thus extending the recent gaussian non-hermitian literature. We develop the general formalism for calculating the Green function and averaged density of eigenvalues, which may be thought of as the non-hermitian analog of the method due to Brèzin, Itzykson, Parisi and Zuber for analyzing hermitian non-gaussian random matrices. We obtain an explicit algebraic equation for the integrated density of eigenvalues. A somewhat surprising result of that equation is that the shape of the eigenvalue distribution in the complex plane is either a disk or an annulus. As a concrete example, we analyze the quartic ensemble and study the phase transition from a disk shaped eigenvalue distribution to an annular distribution. Finally, we apply the method of hermitization to develop the addition formalism for free non-hermitian random variables. We use this formalism to state and prove a non-abelian non-hermitian version of the central limit theorem.

Original language | English |
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Pages (from-to) | 643-669 |

Number of pages | 27 |

Journal | Nuclear Physics B |

Volume | 501 |

Issue number | 3 |

DOIs | |

State | Published - 22 Sep 1997 |

Externally published | Yes |

## ASJC Scopus subject areas

- Nuclear and High Energy Physics