From integrable to chaotic systems: Universal local statistics of Lyapunov exponents

@article{Akemann2018FromIT,
  title={From integrable to chaotic systems: Universal local statistics of Lyapunov exponents},
  author={G. Akemann and Z. Burda and M. Kieburg},
  journal={EPL},
  year={2018},
  volume={126},
  pages={40001}
}
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 $N\times N$ 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… Expand

Figures from this paper

Phase transitions for infinite products of large non-Hermitian random matrices
Products of $M$ i.i.d. non-Hermitian random matrices of size $N \times N$ relate Gaussian fluctuation of Lyapunov and stability exponents in dynamical systems (finite $N$ and large $M$) to localExpand
Lyapunov exponent, universality and phase transition for products of random matrices
We solve the problem on local statistics of finite Lyapunov exponents for $M$ products of $N\times N$ Gaussian random matrices as both $M$ and $N$ go to infinity, proposed by Akemann, Burda, KieburgExpand
Gaussian fluctuations for products of random matrices
We study global fluctuations for singular values of $M$-fold products of several right-unitarily invariant $N \times N$ random matrix ensembles. As $N \to \infty$, we show the fluctuations of theirExpand
Limits and fluctuations of $p$-adic random matrix products.
We show that singular numbers (also known as invariant factors or Smith normal forms) of products and corners of random matrices over $\mathbb{Q}_p$ are governed by the Hall-Littlewood polynomials,Expand
Lyapunov exponents for truncated unitary and Ginibre matrices
In this note, we show that the Lyapunov exponents of mixed products of random truncated Haar unitary and complex Ginibre matrices are asymptotically given by equally spaced ‘picket-fence’ statistics.Expand
Products of Many Large Random Matrices and Gradients in Deep Neural Networks
We study products of random matrices in the regime where the number of terms and the size of the matrices simultaneously tend to infinity. Our main theorem is that the logarithm of the $$\ell _2$$ ℓExpand
Local tail statistics of heavy-tailed random matrix ensembles with unitary invariance
We study heavy-tailed Hermitian random matrices that are unitarily invariant. The invariance implies that the eigenvalue and eigenvector statistics are decoupled. The motivating question has beenExpand
Random Neural Networks in the Infinite Width Limit as Gaussian Processes
  • B. Hanin
  • Computer Science, Mathematics
  • ArXiv
  • 2021
This article gives a new proof that fully connected neural networks with random weights and biases converge to Gaussian processes in the regime where the input dimension, output dimension, and depthExpand
Fluctuations of $\beta$-Jacobi Product Processes
We study Markov chains formed by squared singular values of products of truncated orthogonal, unitary, symplectic matrices (corresponding to the Dyson index $\beta = 1,2,4$ respectively) where timeExpand
PR ] 3 O ct 2 01 9 FLUCTUATIONS OF β-JACOBI PRODUCT PROCESSES
We study Markov chains formed by squared singular values of products of truncated orthogonal, unitary, symplectic matrices (corresponding to the Dyson index β = 1, 2, 4 respectively) where timeExpand
...
1
2
...

References

SHOWING 1-10 OF 72 REFERENCES
Lyapunov exponent, universality and phase transition for products of random matrices
We solve the problem on local statistics of finite Lyapunov exponents for $M$ products of $N\times N$ Gaussian random matrices as both $M$ and $N$ go to infinity, proposed by Akemann, Burda, KieburgExpand
Universal distribution of Lyapunov exponents for products of Ginibre matrices
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 theExpand
Universality in chaos: Lyapunov spectrum and random matrix theory.
TLDR
The existence of a new universality in classical chaotic systems when the number of degrees of freedom is large is proposed: the statistical property of the Lyapunov spectrum is described by random matrix theory and it is demonstrated also in the product of random matrices. Expand
Isotropic Brownian motions over complex fields as a solvable model for May-Wigner stability analysis
We consider matrix-valued stochastic processes known as isotropic Brownian motions, and show that these can be solved exactly over complex fields. While these processes appear in a variety ofExpand
RECENT EXACT AND ASYMPTOTIC RESULTS FOR PRODUCTS OF INDEPENDENT RANDOM MATRICES
In this review we summarise recent results for the complex eigenvalues and singular values of finite products of finite size random matrices, their correlation functions and asymptotic limits. TheExpand
RANDOM-MATRIX THEORIES IN QUANTUM PHYSICS : COMMON CONCEPTS
We review the development of random-matrix theory (RMT) during the last fifteen years. We emphasize both the theoretical aspects, and the application of the theory to a number of fields. TheseExpand
Singular values of products of random matrices and polynomial ensembles
Akemann, Ipsen, and Kieburg showed recently that the squared singular values of a product of M complex Ginibre matrices are distributed according to a determinantal point process. We introduce theExpand
Random graph states, maximal flow and Fuss-Catalan distributions
For any graph consisting of k vertices and m edges we construct an ensemble of random pure quantum states which describe a system composed of 2m subsystems. Each edge of the graph represents aExpand
Products of random matrices in statistical physics
I Background.- 1. Why Study Random Matrices?.- 1.1 Statistics of the Eigenvalues of Random Matrices.- 1.1.1 Nuclear Physics.- 1.1.2 Stability of Large Ecosystems.- 1.1.3 Disordered Harmonic Solids.-Expand
Singular value correlation functions for products of Wishart random matrices
TLDR
This paper first compute the joint probability distribution for the singular values of the product matrix when the matrix size N and the number M are fixed but arbitrary, which leads to a determinantal point process which can be realized in two different ways. Expand
...
1
2
3
4
5
...