Universality of local spectral statistics of products of random matrices

Akemann G, Burda Z, Kieburg M (2020)
Physical Review E 102(5): 052134.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Akemann, GernotUniBi; Burda, Zdzislaw; Kieburg, Mario
Abstract / Bemerkung
We derive exact analytical expressions for correlation functions of singular values of the product of M Ginibre matrices of size N in the double scaling limit M, N -> infinity. The singular value statistics is described by a determinantal point process with a kernel that interpolates between Gaussian unitary ensemble statistic and Dirac-delta (picket-fence) statistic. In the thermodynamic limit N -> infinity, the interpolation parameter is given by the limiting quotient a = N/M. One of our goals is to find an explicit form of the kernel at the hard edge, in the bulk, and at the soft edge for any a. We find that in addition to the standard scaling regimes, there is a transitional regime which interpolates between the hard edge and the bulk. We conjecture that these results are universal, and that they apply to a broad class of products of random matrices from the Gaussian basin of attraction, including correlated matrices. We corroborate this conjecture by numerical simulations. Additionally, we show that the local spectral statistics of the considered random matrix products is identical with the local statistics of Dyson Brownian motion with the initial condition given by equidistant positions, with the crucial difference that this equivalence holds only locally. Finally, we have identified a mesoscopic spectral scale at the soft edge which is crucial for the unfolding of the spectrum.
Erscheinungsjahr
2020
Zeitschriftentitel
Physical Review E
Band
102
Ausgabe
5
Art.-Nr.
052134
ISSN
2470-0045
eISSN
2470-0053
Page URI
https://pub.uni-bielefeld.de/record/2949705

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Akemann G, Burda Z, Kieburg M. Universality of local spectral statistics of products of random matrices. Physical Review E. 2020;102(5): 052134.
Akemann, G., Burda, Z., & Kieburg, M. (2020). Universality of local spectral statistics of products of random matrices. Physical Review E, 102(5), 052134. doi:10.1103/PhysRevE.102.052134
Akemann, G., Burda, Z., and Kieburg, M. (2020). Universality of local spectral statistics of products of random matrices. Physical Review E 102:052134.
Akemann, G., Burda, Z., & Kieburg, M., 2020. Universality of local spectral statistics of products of random matrices. Physical Review E, 102(5): 052134.
G. Akemann, Z. Burda, and M. Kieburg, “Universality of local spectral statistics of products of random matrices”, Physical Review E, vol. 102, 2020, : 052134.
Akemann, G., Burda, Z., Kieburg, M.: Universality of local spectral statistics of products of random matrices. Physical Review E. 102, : 052134 (2020).
Akemann, Gernot, Burda, Zdzislaw, and Kieburg, Mario. “Universality of local spectral statistics of products of random matrices”. Physical Review E 102.5 (2020): 052134.

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PMID: 33327167
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arXiv: 2008.11470

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