Universal distribution of Lyapunov exponents for products of Ginibre matrices
Jun 3, 201435 pages
Published in:
- J.Phys.A 47 (2014) 39, 395202
- Published: Sep 16, 2014
e-Print:
- 1406.0803 [math-ph]
View in:
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Abstract: (IOP)
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 to leading order. We discuss a general mechanism for 2 × 2 matrices showing why the singular values and the radii of complex eigenvalues collapse onto the same value in the large-t limit. We thereby rederive Newmanʼs triangular law which has a simple interpretation as the radial density of complex eigenvalues in the circular law, and study the commutativity of the two limits and on the global scale and on the local scale. We show as a mathematical byproduct that a particular asymptotic expansion of a Meijer G-function with large index leads to a Gaussian.Note:
- 36 pages, 6 figures
- random matrix theory
- product matrices
- Lyapunov exponents
References(63)
Figures(6)
- [1]
- [2]
- [3]
- [4]
- [5]
- [6]
- [7]
- [8]
- [9]
- [10]
- [11]
- [12]
- [13]
- [14]
- [15]
- [16]
- [17]
- [18]
- [19]
- [20]
- [21]
- [22]
- [23]
- [24]
- [25]