Universal distribution of Lyapunov exponents for products of Ginibre matrices

Abstract

Starting from exact analytical results on singular values and complex eigenvalues of products of independent Gaussian complex random $N\times N$ matrices also called 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 compute its $t$-dependent variance as well as corrections in a $1/t$ expansion. Originally Lyapunov exponents are defined for 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 $2\times 2$ matrices why the singular values and the radii of complex eigenvalues collapse onto the same value in the large-$t$ limit. Thereby we rederive Newman's 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 $t\to \infty$ and $N\to \infty$ on the global and the local scale. As a mathematical byproduct we show that a particular asymptotic expansion of a Meijer G-function with large index leads to a Gaussian.

36 pages, 6 figures

References in corpus (4)

• On the Asymptotic Spectrum of Products of Independent Random Matrices
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• Eigenvalue density of Wilson loops in 2D SU(N) YM
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• Universality conjecture and results for a model of several coupled positive-definite matrices
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• Random-matrix models of monitored quantum circuits
• Universality of local spectral statistics of products of random matrices
• Eigenvector statistics of the product of Ginibre matrices
• Probability that product of real random matrices have all eigenvalues real tend to 1
• The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$
• Lyapunov exponents and eigenvalues of products of random matrices
• Local Tail Statistics of Heavy-Tailed Random Matrix Ensembles with Unitary Invariance
• Phase transitions for infinite products of large non-Hermitian random matrices
• Gap probability for products of random matrices in the critical regime