article published yes Guillaume Dubach author D5C6A458-10C4-11EA-ABF4-A4B43DDC885E0000-0001-6892-8137 Yuval Peled author LaEr department ISTplus - Postdoctoral Fellowships project We consider words Gi1⋯Gim 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. Institute of Mathematical Statistics2021 eng StatisticsProbability and UncertaintyStatistics and Probability 0091-1798 1904.04312 00068134900000810.1214/20-aop1496 4941886-1916 Dubach, Guillaume, and Yuval Peled. “On Words of Non-Hermitian Random Matrices.” <i>The Annals of Probability</i>. Institute of Mathematical Statistics, 2021. <a href="https://doi.org/10.1214/20-aop1496">https://doi.org/10.1214/20-aop1496</a>. Dubach, G., &#38; Peled, Y. (2021). On words of non-Hermitian random matrices. <i>The Annals of Probability</i>. Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/20-aop1496">https://doi.org/10.1214/20-aop1496</a> Dubach, Guillaume, and Yuval Peled. “On Words of Non-Hermitian Random Matrices.” <i>The Annals of Probability</i>, vol. 49, no. 4, Institute of Mathematical Statistics, 2021, pp. 1886–916, doi:<a href="https://doi.org/10.1214/20-aop1496">10.1214/20-aop1496</a>. G. Dubach and Y. Peled, “On words of non-Hermitian random matrices,” <i>The Annals of Probability</i>, vol. 49, no. 4. Institute of Mathematical Statistics, pp. 1886–1916, 2021. Dubach G, Peled Y. 2021. On words of non-Hermitian random matrices. The Annals of Probability. 49(4), 1886–1916. Dubach G, Peled Y. On words of non-Hermitian random matrices. <i>The Annals of Probability</i>. 2021;49(4):1886-1916. doi:<a href="https://doi.org/10.1214/20-aop1496">10.1214/20-aop1496</a> G. Dubach, Y. Peled, The Annals of Probability 49 (2021) 1886–1916. 152592024-04-03T07:19:42Z2025-09-10T10:13:20Z