### The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$

Akemann G, Byun S-S (2022)
arXiv:2201.07668.

Preprint | Englisch

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Autor*in
Akemann, GernotUniBi; Byun, Sung-Soo
Abstract / Bemerkung
We study the product $P_m$ of $m$ real Ginibre matrices with Gaussian elements of size $N$, which has received renewed interest recently. Its eigenvalues, which are either real or come in complex conjugate pairs, become all real with probability one when $m\to\infty$ at fixed $N$. In this regime the statistics becomes deterministic and the Lyapunov spectrum has been derived long ago. On the other hand, when $N\to\infty$ and $m$ is fixed, it can be expected that away from the origin the same local statistics as for a single real Ginibre ensemble at $m=1$ prevails. Inspired by analogous findings for products of complex Ginibre matrices, we introduce a critical scaling regime when the two parameters are proportional, $m=\alpha N$. We derive the expected number, variance and rescaled density of real eigenvalues in this critical regime. This allows us to interpolate between previous recent results in the above mentioned limits when $\alpha\to\infty$ and $\alpha\to0$, respectively.
Erscheinungsjahr
2022
Zeitschriftentitel
arXiv:2201.07668
Page URI
https://pub.uni-bielefeld.de/record/2963770

### Zitieren

Akemann G, Byun S-S. The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$. arXiv:2201.07668. 2022.
Akemann, G., & Byun, S. - S. (2022). The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$. arXiv:2201.07668
Akemann, G., and Byun, S. - S. (2022). The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$. arXiv:2201.07668.
Akemann, G., & Byun, S.-S., 2022. The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$. arXiv:2201.07668.
G. Akemann and S.-S. Byun, “The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$”, arXiv:2201.07668, 2022.
Akemann, G., Byun, S.-S.: The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$. arXiv:2201.07668. (2022).
Akemann, Gernot, and Byun, Sung-Soo. “The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$”. arXiv:2201.07668 (2022).

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### Quellen

arXiv: 2201.07668