article published yes László Erdös author 4DBD5372-F248-11E8-B48F-1D18A9856A870000-0001-5366-9603 Hong Chang Ji author dd216c0a-c1f9-11eb-beaf-e9ea9d2de76d LaEr department Random matrices beyond Wigner-Dyson-Mehta project For large dimensional non-Hermitian random matrices X with real or complex independent, identically distributed, centered entries, we consider the fluctuations of f (X) as a matrix where f is an analytic function around the spectrum of X. We prove that for a generic bounded square matrix A, the quantity Tr f (X)A exhibits Gaussian fluctuations as the matrix size grows to infinity, which consists of two independent modes corresponding to the tracial and traceless parts of A. We find a new formula for the variance of the traceless part that involves the Frobenius norm of A and the L2-norm of f on the boundary of the limiting spectrum. On étudie les fluctuations de f (X), où X est une matrice aléatoire non-hermitienne de grande taille à coefficients i.i.d. (réels ou complexes), et f une fonction analytique sur un domaine qui contient le spectre de X. On prouve que, pour une matrice carrée générique et bornée A, les fluctuations de la quantité tr f (X)A sont asymptotiquement gaussiennes et comportent deux modes indépendants, correspondant aux composantes traciale et de trace nulle de A. Une nouvelle formule est établie pour la variance de la composante de trace nulle, qui fait intervenir la norme de Frobenius de A et la norme L2 de f sur la frontière du spectre limite. Institute of Mathematical Statistics2023 eng 0246-0203 2112.11382 00109845640001010.1214/22-AIHP1304 5942083-2105 Erdös L, Ji HC. 2023. Functional CLT for non-Hermitian random matrices. Annales de l’institut Henri Poincare (B) Probability and Statistics. 59(4), 2083–2105. L. Erdös, H.C. Ji, Annales de l’institut Henri Poincare (B) Probability and Statistics 59 (2023) 2083–2105. Erdös, L., &#38; Ji, H. C. (2023). Functional CLT for non-Hermitian random matrices. <i>Annales de l’institut Henri Poincare (B) Probability and Statistics</i>. Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/22-AIHP1304">https://doi.org/10.1214/22-AIHP1304</a> Erdös, László, and Hong Chang Ji. “Functional CLT for Non-Hermitian Random Matrices.” <i>Annales de l’institut Henri Poincare (B) Probability and Statistics</i>. Institute of Mathematical Statistics, 2023. <a href="https://doi.org/10.1214/22-AIHP1304">https://doi.org/10.1214/22-AIHP1304</a>. Erdös L, Ji HC. Functional CLT for non-Hermitian random matrices. <i>Annales de l’institut Henri Poincare (B) Probability and Statistics</i>. 2023;59(4):2083-2105. doi:<a href="https://doi.org/10.1214/22-AIHP1304">10.1214/22-AIHP1304</a> L. Erdös and H. C. Ji, “Functional CLT for non-Hermitian random matrices,” <i>Annales de l’institut Henri Poincare (B) Probability and Statistics</i>, vol. 59, no. 4. Institute of Mathematical Statistics, pp. 2083–2105, 2023. Erdös, László, and Hong Chang Ji. “Functional CLT for Non-Hermitian Random Matrices.” <i>Annales de l’institut Henri Poincare (B) Probability and Statistics</i>, vol. 59, no. 4, Institute of Mathematical Statistics, 2023, pp. 2083–105, doi:<a href="https://doi.org/10.1214/22-AIHP1304">10.1214/22-AIHP1304</a>. 146672023-12-10T23:01:00Z2025-09-09T13:41:08Z