Journal article

Universal distribution of Lyapunov exponents for products of Ginibre matrices

Gernot Akemann, Zdzislaw Burda, Mario Kieburg

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL | IOP PUBLISHING LTD | Published : 2014

Abstract

Starting from exact analytical results on singular values and complex eigenvalues of products of independent Gaussian complex random N N matrices, also called the Ginibre ensemble, we rederive the Lyapunov exponents for an infinite product. We show that for a large number t of product matrices, the distribution of each Lyapunov exponent is normal, and we compute its t-dependent variance as well as corrections in a large-t expansion. Originally Lyapunov exponents are defined for the singular values of the product matrix that represents a linear time evolution. Surprisingly a similar construction for the moduli of the complex eigenvalues yields the very same exponents and normal distributions ..

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University of Melbourne Researchers

Grants

Awarded by German research council DFG


Awarded by National Centre of Science in Poland


Funding Acknowledgements

We would like to acknowledge SFB vertical bar TR 12 'Symmetries and Universality in Mesoscopic Systems' of the German research council DFG for partial support (GA). ZB was supported by the Alexander von Humboldt Foundation and the Grant DEC-2011/02/A/ST1/00119 of the National Centre of Science in Poland, and MK was supported by a Feodor Lynen return fellowship of the Alexander von Humboldt Foundation. We also thank Jesper R Ipsen as well as Jens Markloff for fruitful discussions.