article
Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices
published
yes
Giorgio
Cipolloni
author 42198EFA-F248-11E8-B48F-1D18A9856A870000-0002-4901-7992
László
Erdös
author 4DBD5372-F248-11E8-B48F-1D18A9856A870000-0001-5366-9603
Dominik J
Schröder
author 408ED176-F248-11E8-B48F-1D18A9856A870000-0002-2904-1856
LaEr
department
Random matrices, universality and disordered quantum systems
project
International IST Doctoral Program
project
We consider large non-Hermitian random matrices X with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having 2+ϵ derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [72], or the distribution of the matrix elements needed to be Gaussian [73], or at least match the Gaussian up to the first four moments [82, 56]. We find the exact dependence of the limiting variance on the fourth cumulant that was not known before. The proof relies on two novel ingredients: (i) a local law for a product of two resolvents of the Hermitisation of X with different spectral parameters and (ii) a coupling of several weakly dependent Dyson Brownian motions. These methods are also the key inputs for our analogous results on the linear eigenvalue statistics of real matrices X that are presented in the companion paper [32].
https://research-explorer.ista.ac.at/download/10405/14388/2023_CommPureMathematics_Cipolloni.pdf
application/pdfno
Wiley2023
eng
Communications on Pure and Applied Mathematics
0010-3640
1097-0312
1912.04100
00072465250000110.1002/cpa.22028
765946-1034
G. Cipolloni, L. Erdös, D.J. Schröder, Communications on Pure and Applied Mathematics 76 (2023) 946–1034.
Cipolloni, Giorgio, et al. “Central Limit Theorem for Linear Eigenvalue Statistics of Non-Hermitian Random Matrices.” <i>Communications on Pure and Applied Mathematics</i>, vol. 76, no. 5, Wiley, 2023, pp. 946–1034, doi:<a href="https://doi.org/10.1002/cpa.22028">10.1002/cpa.22028</a>.
G. Cipolloni, L. Erdös, and D. J. Schröder, “Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices,” <i>Communications on Pure and Applied Mathematics</i>, vol. 76, no. 5. Wiley, pp. 946–1034, 2023.
Cipolloni, G., Erdös, L., & Schröder, D. J. (2023). Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices. <i>Communications on Pure and Applied Mathematics</i>. Wiley. <a href="https://doi.org/10.1002/cpa.22028">https://doi.org/10.1002/cpa.22028</a>
Cipolloni G, Erdös L, Schröder DJ. Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices. <i>Communications on Pure and Applied Mathematics</i>. 2023;76(5):946-1034. doi:<a href="https://doi.org/10.1002/cpa.22028">10.1002/cpa.22028</a>
Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Central Limit Theorem for Linear Eigenvalue Statistics of Non-Hermitian Random Matrices.” <i>Communications on Pure and Applied Mathematics</i>. Wiley, 2023. <a href="https://doi.org/10.1002/cpa.22028">https://doi.org/10.1002/cpa.22028</a>.
Cipolloni G, Erdös L, Schröder DJ. 2023. Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices. Communications on Pure and Applied Mathematics. 76(5), 946–1034.
104052021-12-05T23:01:41Z2023-10-04T09:22:55Z