### Products of random matrices from fixed trace and induced Ginibre ensembles

Akemann G, Cikovic M (2018)
Journal of Physics. A, Mathematical and Theoretical 51(18): 34.

Zeitschriftenaufsatz | Veröffentlicht | Englisch

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Abstract / Bemerkung
We investigate the microcanonical version of the complex induced Ginibre ensemble, by introducing a fixed trace constraint for its second moment. Like for the canonical Ginibre ensemble, its complex eigenvalues can be interpreted as a two-dimensional Coulomb gas, which are now subject to a constraint and a modified, collective confining potential. Despite the lack of determinantal structure in this fixed trace ensemble, we compute all its density correlation functions at finite matrix size and compare to a fixed trace ensemble of normal matrices, representing a different Coulomb gas. Our main tool of investigation is the Laplace transform, that maps back the fixed trace to the induced Ginibre ensemble. Products of random matrices have been used to study the Lyapunov and stability exponents for chaotic dynamical systems, where the latter are based on the complex eigenvalues of the product matrix. Because little is known about the universality of the eigenvalue distribution of such product matrices, we then study the product of m induced Ginibre matrices with a fixed trace constraint-which are clearly non-Gaussian-and M - m such Ginibre matrices without constraint. Using an m-fold inverse Laplace transform, we obtain a concise result for the spectral density of such a mixed product matrix at finite matrix size, for arbitrary fixed m and M. Very recently local and global universality was proven by the authors and their coworker for a more general, single elliptic fixed trace ensemble in the bulk of the spectrum. Here, we argue that the spectral density of mixed products is in the same universality class as the product of M independent induced Ginibre ensembles.
Stichworte
products of random matrices; fixed trace; universality
Erscheinungsjahr
2018
Zeitschriftentitel
Journal of Physics. A, Mathematical and Theoretical
Band
51
Ausgabe
18
Art.-Nr.
34
ISSN
1751-8113
eISSN
1751-8121
Page URI
https://pub.uni-bielefeld.de/record/2920257

### Zitieren

Akemann G, Cikovic M. Products of random matrices from fixed trace and induced Ginibre ensembles. Journal of Physics. A, Mathematical and Theoretical. 2018;51(18): 34.
Akemann, G., & Cikovic, M. (2018). Products of random matrices from fixed trace and induced Ginibre ensembles. Journal of Physics. A, Mathematical and Theoretical, 51(18), 34. doi:10.1088/1751-8121/aab8a9
Akemann, G., and Cikovic, M. (2018). Products of random matrices from fixed trace and induced Ginibre ensembles. Journal of Physics. A, Mathematical and Theoretical 51:34.
Akemann, G., & Cikovic, M., 2018. Products of random matrices from fixed trace and induced Ginibre ensembles. Journal of Physics. A, Mathematical and Theoretical, 51(18): 34.
G. Akemann and M. Cikovic, “Products of random matrices from fixed trace and induced Ginibre ensembles”, Journal of Physics. A, Mathematical and Theoretical, vol. 51, 2018, : 34.
Akemann, G., Cikovic, M.: Products of random matrices from fixed trace and induced Ginibre ensembles. Journal of Physics. A, Mathematical and Theoretical. 51, : 34 (2018).
Akemann, Gernot, and Cikovic, Milan. “Products of random matrices from fixed trace and induced Ginibre ensembles”. Journal of Physics. A, Mathematical and Theoretical 51.18 (2018): 34.

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### Quellen

arXiv: 1711.05488