Journal article
From integrable to chaotic systems: Universal local statistics of Lyapunov exponents
Mario Kieburg, Burda Akemann, Zdzsilaw Burda
EPL (Europhysics Letters) | IOP Publishing | Published : 2019
Abstract
Systems where time evolution follows a multiplicative process are ubiquitous in physics. We study a toy model for such systems where each time step is given by multiplication with an independent random N N matrix with complex Gaussian elements, the complex Ginibre ensemble. This model allows to explicitly compute the Lyapunov exponents and local correlations amongst them, when the number of factors M becomes large. While the smallest eigenvalues always remain deterministic, which is also the case for many chaotic quantum systems, we identify a critical double scaling limit N ∼M for the rest of the spectrum. It interpolates between the known deterministic behaviour of the Lyapunov exponents f..
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Awarded by German research council (DFG)
Awarded by AGH UST of the Ministry of Science and Higher Education
Funding Acknowledgements
We acknowledge support by the German research council (DFG) via CRC 1283: "Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications" (GA, MK) and by the AGH UST statutory tasks No. 11.11.220.01/2 within subsidy of the Ministry of Science and Higher Education (ZB). We thank D.-Z. LIU and D. WANG for sharing their results about the same double scaling limit at the soft edge derived independently [48], and are grateful for D.-Z. Liu's comments on a preprint version of this letter. Moreover, we are indebted to M. Duits for pointing out a possible relation to [45], and to J. J. M. Verbaarschot for confirming this for the microscopic density in the bulk, by applying the Poisson summation formula. Finally, we also appreciate fruitful discussions with J. Ipsen on the DMPK equation.