Lyapunov Exponents for Some Isotropic Random Matrix Ensembles

Forrester, Peter J.; Zhang, Jiyuan

doi:10.1007/s10955-019-02474-2

Cited by 4 publications

(2 citation statements)

References 37 publications

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“…On the other hand, as expected from the structure (6.5), the analysis for the radial density is considerably simplified; see [111,115,11]. We also mention that in the same spirit as Remark 4.13, the Lyapunov and stability exponents of the products of GinSE are available in the literature [120,112,94].…”

Section: Truncations Of Haar Real Quaternion Symplectic Matricesmentioning

confidence: 79%

“…For GinOE matrices, the explicit Lyapunov spectrum was computed long ago by Newman [106]. This exact result in the case of the largest Lyapunov exponent was generalised in [120,94] to allow for the GinOE matrices to be left multiplied by a positive definite matrix. A direct study of the stability exponents associated with the modulus of the eigenvalues -these becoming all real as M → ∞ -was undertaken in [113].…”

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Progress on the study of the Ginibre ensembles II: GinOE and GinSE

Byun¹,

Forrester²

2023

Preprint

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This is part II of a review relating to the three classes of random non-Hermitian Gaussian matrices introduced by Ginibre in 1965. While part I restricted attention to the GinUE (Ginibre unitary ensemble) case of complex elements, in this part the cases of real elements (GinOE, denoting Ginibre orthogonal ensemble) and quaternion elements represented as 2 × 2 complex blocks (GinSE, denoting Ginibre symplectic ensemble) are considered. The eigenvalues of both GinOE and GinSE form Pfaffian point processes, which are more complicated than the determinantal point processes resulting from GinUE. Nevertheless, many of the obstacles that have slowed progress on the development of traditional aspects of the theory have now been overcome, while new theoretical aspects and new applications have been identified. This permits a comprehensive account of themes addressed too in the complex case: eigenvalue probability density functions and correlation functions, limit formulas for correlation functions, fluctuation formulas, sum rules, gap probabilities and eigenvector statistics, among others. Distinct from the complex case is the need to develop a theory of skew orthogonal polynomials corresponding to the skew inner product associated with the Pfaffian. Another distinct theme is the statistics of real eigenvalues, which is unique to GinOE. These appear in a number of applications of the theory, coming from areas as diverse as diffusion processes and persistence in statistical physics, topologically driven parametric energy level crossings for certain quantum dots, and equilibria counting for a system of random nonlinear differential equations.

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Section: Truncations Of Haar Real Quaternion Symplectic Matricesmentioning

confidence: 79%

mentioning

confidence: 99%

Progress on the study of the Ginibre ensembles II: GinOE and GinSE

Byun¹,

Forrester²

2023

Preprint

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Discrete matrix hypergeometric functions

Cuchta

Grow²,

Wintz³

2023

Journal of Mathematical Analysis and Applications

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CLT with explicit variance for products of random singular matrices related to Hill’s equation

Mariano

Panzo

2021

Random Matrices: Theory Appl.

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We prove a central limit theorem (CLT) for the product of a class of random singular matrices related to a random Hill’s equation studied by Adams–Bloch–Lagarias. The CLT features an explicit formula for the variance in terms of the distribution of the matrix entries and this allows for exact calculation in some examples. Our proof relies on a novel connection to the theory of [Formula: see text]-dependent sequences which also leads to an interesting and precise nondegeneracy condition.

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Lyapunov Exponents for Some Isotropic Random Matrix Ensembles

Cited by 4 publications

References 37 publications

Progress on the study of the Ginibre ensembles II: GinOE and GinSE

Progress on the study of the Ginibre ensembles II: GinOE and GinSE

Discrete matrix hypergeometric functions

CLT with explicit variance for products of random singular matrices related to Hill’s equation

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