Asymptotics of finite system Lyapunov exponents for some random matrix ensembles

Abstract

For products $P_N$ of $N$ random matrices of size $d\times d$, there is a natural notion of finite $N$ Lyapunov exponents $\{{\mu }_i{\}}_{i=1}^d$. In the case of standard Gaussian random matrices with real, complex or real quaternion elements, and extended to the general variance case for ${\mu }_1$, methods known for the computation of ${\lim }_{N\to \infty }⟨{\mu }_i⟩$ are used to compute the large $N$ form of the variances of the exponents. Analogous calculations are performed in the case that the matrices making up $P_N$ are products of sub-blocks of random unitary matrices with Haar measure. Furthermore, we make some remarks relating to the coincidence of the Lyapunov exponents and the stability exponents relating to the eigenvalues of $P_N$.

arXiv comment15 pages

References in corpus (8)

• Truncations of random unitary matrices
• Weak Commutation Relations and Eigenvalue Statistics for Products of Rectangular Random Matrices
• Capacity and Character Expansions: Moment generating function and other exact results for MIMO correlated channels
• Infinite Products of Large Random Matrices and Matrix-valued Diffusion
• Products of Independent Quaternion Ginibre Matrices and their Correlation Functions
• Lyapunov exponents for products of complex Gaussian random matrices
• On the largest Lyapunov exponent for products of Gaussian matrices
• Lyapunov exponents for products of rectangular real, complex and quaternionic Ginibre matrices

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