PUB - Publikationen an der Universität Bielefeld

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.