# On Words of non-Hermitian Random Matrices

## TL;DR

This paper investigates the spectral properties of words formed from non-Hermitian random matrices, revealing that their squared singular values follow Fuss-Catalan distributions and establishing a topological framework for their fluctuations.

## Contribution

It generalizes known results on powers and products of Ginibre matrices to arbitrary words, introduces combinatorial parameters like coperiod, and extends findings to broader classes of matrices under moment-matching assumptions.

## Key findings

- Squared singular values follow Fuss-Catalan distributions.
- Coperiod characterizes eigenvalue fluctuations.
- Central limit theorem established for traces of matrix words.

## Abstract

We consider words $G_{i_1} \cdots G_{i_m}$ 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.