# 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): 395202.

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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):395202.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), 395202. doi:10.1088/1751-8113/47/39/395202Akemann, 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, 395202.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), p 395202. 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, pp. 395202. Akemann, G., Burda, Z., Kieburg, M.: Universal distribution of Lyapunov exponents for products of Ginibre matrices. Journal of Physics A: Mathematical and Theoretical. 47, 395202 (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): 395202.
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