On the Largest Lyapunov Exponent for Products of Gaussian Matrices

Kargin, Vladislav

doi:10.1007/s10955-014-1077-9

Cited by 24 publications

(39 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…. , n. In the past few years, new interest has arisen in this limit due to explicit results for the joint densities of the singular value and the eigenvalues at finite n and M , see [4,24,26,36,40,55]. In particular, the very recent work [55] contains a general result on the Lyapunov and stability exponents which we cite here.…”

Section: Central Limit Theorem For Lyapunov Exponentsmentioning

confidence: 99%

“…[57], [15] and the references therein. Interest in this area has resurged more recently due to explicit results for finite products [4,24,26,36,40,55]. In particular, in [55] it is shown that, under certain conditions, for products of independently and identically distributed, bi-unitarily invariant random matrices of fixed dimension, the logarithms of the singular values and the complex eigenvalues are asymptotically Gaussian distributed as the number of factors tends to infinity.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Products of random matrices from polynomial ensembles

Kieburg¹,

Kösters²

2019

Ann. Inst. H. Poincaré Probab. Statist.

View full text Add to dashboard Cite

Very recently we have shown that the spherical transform is a convenient tool for studying the relation between the joint density of the singular values and that of the eigenvalues for bi-unitarily invariant random matrices. In the present work we discuss the implications of these results for products of random matrices. In particular, we derive a transformation formula for the joint densities of a product of two independent bi-unitarily invariant random matrices, the first from a polynomial ensemble and the second from a polynomial ensemble of derivative type. This allows us to re-derive and generalize a number of recent results in random matrix theory, including a transformation formula for the kernels of the corresponding determinantal point processes. Starting from these results, we construct a continuous family of random matrix ensembles interpolating between the products of different numbers of Ginibre matrices and inverse Ginibre matrices. Furthermore, we make contact to the asymptotic distribution of the Lyapunov exponents of the products of a large number of bi-unitarily invariant random matrices of fixed dimension.

show abstract

Section: Central Limit Theorem For Lyapunov Exponentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Products of random matrices from polynomial ensembles

Kieburg¹,

Kösters²

2019

Ann. Inst. H. Poincaré Probab. Statist.

View full text Add to dashboard Cite

show abstract

“…Here, it is assumed that a i = a j for i = j. The similarity between (39) and (37) is immediately recognised. Thus the Lyapunov exponents in (37) may be reinterpreted as equidistantly spaced eigenvalues of an external source matrix, as considered e.g.…”

Section: Solving the Fokker-planck Equation Over Complex Fieldsmentioning

confidence: 99%

Isotropic Brownian motions over complex fields as a solvable model for May–Wigner stability analysis

Ipsen¹,

Schomerus²

2016

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

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 of questions in mathematical physics, our main motivation is their relation to a May-Wigner-like stability analysis, for which we obtain a stability phase diagram. The exact results establish the full joint probability distribution of the finite-time Lyapunov exponents, and may be used as a starting point for a more detailed analysis of the stability-instability phase transition. Our derivations rest on an explicit formulation of a Fokker-Planck equation for the Lyapunov exponents. This formulation happens to coincide with an exactly solvable class of models of the Calgero-Sutherland type, originally encountered for a model of phase-coherent transport. The exact solution over complex fields describes a determinantal point process of biorthogonal type similar to recent results for products of random matrices, and is also closely related to Hermitian matrix models with an external source.

show abstract

“…This has been derived previously in [17] (the additional factor of β in the logarithm on the LHS of (3.13) is due to our use of standard real Gaussians, whereas in [17], standard complex (quaternion) Gaussians are used for β = 2 (β = 4)). With S N defined above (3.6), another viewpoint on (3.13) is that to leading order for large N,…”

Section: The Gaussian Casementioning

confidence: 96%

“…For β = 2 an evaluation easily seen to be equivalent to (3.12) with β = 2 is also given in [17,Eq. (2)].…”

Section: The Gaussian Casementioning

confidence: 99%

Lyapunov Exponents for Some Isotropic Random Matrix Ensembles

Forrester

Zhang

2020

J Stat Phys

View full text Add to dashboard Cite

A random matrix with rows distributed as a function of their length is said to be isotropic. When these distributions are Gaussian, beta type I, or beta type II, previous work has, from the viewpoint of integral geometry, obtained the explicit form of the distribution of the determinant. We use these result to evaluate the sum of the Lyapunov spectrum of the corresponding random matrix product, and we further give explicit expressions for the largest Lyapunov exponent. Generalisations to the case of complex or quaternion entries are also given. For standard Gaussian matrices X, the full Lyapunov spectrum for products of random matrices I N + 1 c X is computed in terms of a generalised hypergeometric function in general, and in terms of a single single integral involving a modified Bessel function for the largest Lyapunov exponent.

show abstract

On the Largest Lyapunov Exponent for Products of Gaussian Matrices

Cited by 24 publications

References 20 publications

Products of random matrices from polynomial ensembles

Products of random matrices from polynomial ensembles

Isotropic Brownian motions over complex fields as a solvable model for May–Wigner stability analysis

Lyapunov Exponents for Some Isotropic Random Matrix Ensembles

Contact Info

Product

Resources

About