## Abstract

We consider words G_{i}_{1} … G_{i}_{m} involving i.i.d. complex Ginibre matrices and study tracial expressions of their eigenvalues and singular values. We show that the limit distribution of the squared singular values of every word of length m is a Fuss–Catalan distribution with parameter m + 1. This generalizes previous results concerning powers of a complex Ginibre matrix and products of independent Ginibre matrices. In addition, we find other combinatorial parameters of the word that determine the second-order limits of the spectral statistics. For instance, the so-called coperiod of a word characterizes the fluctuations of the eigenvalues. We extend these results to words of general non-Hermitian matrices with i.i.d. entries under moment-matching assumptions, band matrices, and sparse matrices. These results rely on the moments method and genus expansion, relating Gaussian matrix integrals to the counting of compact orientable surfaces of a given genus. This allows us to derive a central limit theorem for the trace of any word of complex Ginibre matrices and their conjugate transposes, where all parameters are defined topologically.

Original language | English |
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Pages (from-to) | 1886-1916 |

Number of pages | 31 |

Journal | Annals of Probability |

Volume | 49 |

Issue number | 4 |

DOIs | |

State | Published - Jul 2021 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© Institute of Mathematical Statistics, 2021

## Keywords

- Complex Ginibre ensemble
- Fuss–Catalan distribution
- genus expansion
- mixed moments of non-Hermitian matrices
- second order freeness
- words of random matrices