Universal distribution of Lyapunov exponents for products of Ginibre matrices

Akemann G, Burda Z, Kieburg M (2014)
Journal of Physics A: Mathematical and Theoretical 47(39).

Journal Article | Published | English

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Abstract
Starting from exact analytical results on singular values and complexeigenvalues of products of independent Gaussian complex random $N\times N$matrices also called Ginibre ensemble we rederive the Lyapunov exponents for aninfinite product. We show that for a large number $t$ of product matrices thedistribution of each Lyapunov exponent is normal and compute its $t$-dependentvariance as well as corrections in a $1/t$ expansion. Originally Lyapunovexponents are defined for singular values of the product matrix that representsa linear time evolution. Surprisingly a similar construction for the moduli ofthe complex eigenvalues yields the very same exponents and normal distributionsto leading order. We discuss a general mechanism for $2\times 2$ matrices whythe singular values and the radii of complex eigenvalues collapse onto the samevalue in the large-$t$ limit. Thereby we rederive Newman's triangular law whichhas a simple interpretation as the radial density of complex eigenvalues in thecircular law and study the commutativity of the two limits $t\to\infty$ and$N\to\infty$ on the global and the local scale. As a mathematical byproduct weshow that a particular asymptotic expansion of a Meijer G-function with largeindex leads to a Gaussian.
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Akemann G, Burda Z, Kieburg M. Universal distribution of Lyapunov exponents for products of Ginibre matrices. Journal of Physics A: Mathematical and Theoretical. 2014;47(39).
Akemann, G., Burda, Z., & Kieburg, M. (2014). Universal distribution of Lyapunov exponents for products of Ginibre matrices. Journal of Physics A: Mathematical and Theoretical, 47(39).
Akemann, G., Burda, Z., and Kieburg, M. (2014). Universal distribution of Lyapunov exponents for products of Ginibre matrices. Journal of Physics A: Mathematical and Theoretical 47.
Akemann, G., Burda, Z., & Kieburg, M., 2014. Universal distribution of Lyapunov exponents for products of Ginibre matrices. Journal of Physics A: Mathematical and Theoretical, 47(39).
G. Akemann, Z. Burda, and M. Kieburg, “Universal distribution of Lyapunov exponents for products of Ginibre matrices”, Journal of Physics A: Mathematical and Theoretical, vol. 47, 2014.
Akemann, G., Burda, Z., Kieburg, M.: Universal distribution of Lyapunov exponents for products of Ginibre matrices. Journal of Physics A: Mathematical and Theoretical. 47, (2014).
Akemann, Gernot, Burda, Zdzislaw, and Kieburg, Mario. “Universal distribution of Lyapunov exponents for products of Ginibre matrices”. Journal of Physics A: Mathematical and Theoretical 47.39 (2014).
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