Characteristic polynomials of products of non-Hermitian Wigner matrices: finite-N results and Lyapunov universality

Akemann G, Götze F, Neuschel T (2021)
Electronic Communications in Probability 26: 30.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
We compute the average characteristic polynomial of the Hermitised product of M real or complex non-Hermitian Wigner matrices of size N x 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 -> infinity 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 M th root. The position of the j th 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.
Stichworte
averages of characteristic polynomials; products of random matrices; non-Hermitian Wigner matrices; Lyapunov exponents; universality
Erscheinungsjahr
2021
Zeitschriftentitel
Electronic Communications in Probability
Band
26
Art.-Nr.
30
eISSN
1083-589X
Page URI
https://pub.uni-bielefeld.de/record/2955377

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Akemann G, Götze F, Neuschel T. Characteristic polynomials of products of non-Hermitian Wigner matrices: finite-N results and Lyapunov universality. Electronic Communications in Probability . 2021;26: 30.
Akemann, G., Götze, F., & Neuschel, T. (2021). Characteristic polynomials of products of non-Hermitian Wigner matrices: finite-N results and Lyapunov universality. Electronic Communications in Probability , 26, 30. https://doi.org/10.1214/21-ECP398
Akemann, G., Götze, F., and Neuschel, T. (2021). Characteristic polynomials of products of non-Hermitian Wigner matrices: finite-N results and Lyapunov universality. Electronic Communications in Probability 26:30.
Akemann, G., Götze, F., & Neuschel, T., 2021. Characteristic polynomials of products of non-Hermitian Wigner matrices: finite-N results and Lyapunov universality. Electronic Communications in Probability , 26: 30.
G. Akemann, F. Götze, and T. Neuschel, “Characteristic polynomials of products of non-Hermitian Wigner matrices: finite-N results and Lyapunov universality”, Electronic Communications in Probability , vol. 26, 2021, : 30.
Akemann, G., Götze, F., Neuschel, T.: Characteristic polynomials of products of non-Hermitian Wigner matrices: finite-N results and Lyapunov universality. Electronic Communications in Probability . 26, : 30 (2021).
Akemann, Gernot, Götze, Friedrich, and Neuschel, Thorsten. “Characteristic polynomials of products of non-Hermitian Wigner matrices: finite-N results and Lyapunov universality”. Electronic Communications in Probability 26 (2021): 30.

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