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2021 Characteristic polynomials of products of non-Hermitian Wigner matrices: finite-N results and Lyapunov universality
Gernot Akemann, Friedrich Götze, Thorsten Neuschel
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Electron. Commun. Probab. 26: 1-13 (2021). DOI: 10.1214/21-ECP398


We compute the average characteristic polynomial of the Hermitised product of M real or complex non-Hermitian Wigner matrices of size N×N with i.i.d. matrix elements, and the average of the characteristic polynomial of a product of M such matrices times the characteristic polynomial of the conjugate product matrix. Surprisingly, the results agree with that of the product of M real or complex Ginibre matrices at finite-N, which have i.i.d. Gaussian entries. For the latter the average characteristic polynomial yields the orthogonal polynomial for the singular values of the product matrix, whereas the product of the two characteristic polynomials involves the kernel of complex eigenvalues. This extends the result of Forrester and Gamburd for one characteristic polynomial of such a single random matrix and only depends on the first two moments. In the limit M at fixed N we determine the locations of the zeros of a single characteristic polynomial, rescaled as Lyapunov exponents by taking the logarithm of the Mth root. The position of the jth zero agrees asymptotically for large-j with the position of the jth Lyapunov exponent for products of Gaussian random matrices, hinting at the universality of the latter.


Support by the German research council DFG through the grant CRC 1283 “Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications” is acknowldged. We thank Mario Kieburg for useful discussions, and the Department of Mathematics at the Royal Institute of Technology (KTH) Stockholm for hospitality (G.A. and T.N.).


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Gernot Akemann. Friedrich Götze. Thorsten Neuschel. "Characteristic polynomials of products of non-Hermitian Wigner matrices: finite-N results and Lyapunov universality." Electron. Commun. Probab. 26 1 - 13, 2021.


Received: 7 July 2020; Accepted: 27 April 2021; Published: 2021
First available in Project Euclid: 25 May 2021

Digital Object Identifier: 10.1214/21-ECP398

Primary: 60B20
Secondary: 37H15

Keywords: averages of characteristic polynomials , Lyapunov exponents , non-Hermitian Wigner matrices , Products of random matrices , Universality


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