Universal distribution of Lyapunov exponents for products of Ginibre matrices

Akemann, Gernot; Burda, Z.; Kieburg, Mario

doi:10.1088/1751-8113/47/39/395202

Cited by 45 publications

(113 citation statements)

References 80 publications

(180 reference statements)

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“…This result has already been obtained in [2] using the exact densities of singular values and eigenvalues. 2 ) for all i = 1, 2...d. This result about Lyapunov exponents of truncated Haar unitary(orthogonal) matrices has already been obtained in [9] using the exact density of truncated Haar unitary(orthogonal) matrices.…”

Section: Then Bothsupporting

confidence: 68%

“…Stability exponents have been considered first in the setting of dynamical systems in [10] and therein equality of Lyapunov and stability exponents has been conjectured based upon plausible arguments and numerical results. Recent comparative studies ( [2], [13]) of Lyapunov exponents and stability exponents in the case of Ginibre matrix ensembles have verified the conjecture to be true in the respective cases. And also ( [9]) mentions this in the case of random truncated unitary matrices.…”

Section: Definitionmentioning

confidence: 79%

“…For a summary of results on this topic, we refer reader to [4]. In [2], a proof of this conjecture in the case of 2 × 2 isotropic random matrices has been given. In section 3, we give a proof of this conjecture for isotropic random matrices of any order.…”

Section: Definitionmentioning

confidence: 99%

“…But the knowledge of variances of these distributions has been very limited until recently. In [2], asymptotic distributions of first order fluctuations of (logarithm of) both singular values and moduli of eigenvalues for product of large number of Ginibre matrices have been computed and shown to be equal. In [9], variances of asymptotic distributions of first order fluctuations of singular values in cases of generalised Ginibre matrices and truncated Haar unitary matrices have been computed.…”

Section: Definitionmentioning

confidence: 99%

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Equality of Lyapunov and Stability Exponents for Products of Isotropic Random Matrices

Reddy

2017

International Mathematics Research Notices

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Abstract. We show that Lyapunov exponents and stability exponents are equal in the case of product of i.i.d isotropic(also known as bi-unitarily invariant) random matrices. We also derive aysmptotic distribution of singular values and eigenvalues of these product random matrices. Moreover, Lyapunov exponents are distinct, unless the random matrices are random scalar multiples of Haar unitary matrices or orthogonal matrices. As a corollary of above result, we show probability that product of n i.i.d real isotropic random matrices has all eigenvalues real goes to one as n → ∞. Also, in the proof of a lemma, we observe that a real (complex) Ginibre matrix can be written as product of a random lower triangular matrix and an independent truncated Haar orthogonal (unitary) matrix. Definitions and introductionLet M 1 , M 2 , . . . be sequence of i.i.d random matrices of order d. Define σ n to be diagonal matrix with singular values of product matrix P n = M 1 M 2 ..M n in the diagonal in decreasing order and similarly λ n to be diagonal matrix with eigenvalues of P n in the diagonal in decreasing order of absolute values, for n = 1, 2, . . .. Let |λ n | 1 n and |σ n | 1 n denote diagonal matrices with non-negative n-th roots of absolute values of diagonal entries of λ n and σ n in the diagonal, respectively.Define σ := lim n→∞ |σ n | 1 n and λ := lim n→∞ |λ n | 1 n , if the limits exist. Then diagonal elements of ln σ and ln λ are called Lyapunov exponents and stability exponents for products of i.i.d random matrices, respectively. In other words, they are rates of exponential growth(or decay) of singular values and eigenvalues of product matrices P n , respectively as n → ∞.We consider both real and complex random matrices in this paper.For the sake of simplicity, we restrict ourselves mostly to complex random matrices . But all the definitions and statements, along with proofs, carry over immediately to real case (by replacing everywhere unitary matrices by orthogonal matrices). Definition 1. A random matrix M is said to be isotropic if probability distribution of U M V is same as that of M , for all unitary matrices U, V .They also go by the names of bi-unitarily invariant and rotation invariant random matrices. It follows from definition that distribution of U M V is same as that of M , if U, V are Haar distributed random unitary matrices independent of M and each other. M = P DQ be the singular value decomposition of M , then U M V =

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Section: Then Bothsupporting

confidence: 68%

Section: Definitionmentioning

confidence: 79%

Section: Definitionmentioning

confidence: 99%

Section: Definitionmentioning

confidence: 99%

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Equality of Lyapunov and Stability Exponents for Products of Isotropic Random Matrices

Reddy

2017

International Mathematics Research Notices

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“…As mentioned before, the non-hermitian products of random matrices appear in multiple contexts including iterative stochastic evolution of linear systems [52][53][54], capacity of complicated MIMO networks [55,56] or analysis of multidimensional data with asymmetric correlations, like e.g. time series of stock prices [21,22,57,58] and others.…”

Section: Discussionmentioning

confidence: 99%

QuaternionicRtransform and non-Hermitian random matrices

Burda

Święch

2015

Phys. Rev. E

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Using the Cayley-Dickson construction we rephrase and review the non-hermitian diagrammatic formalism [R. A. Janik, M. A. Nowak, G. Papp and I. Zahed, Nucl.Phys. B 501, 603 (1997)], that generalizes the free probability calculus to asymptotically large non-hermitian random matrices. The main object in this generalization is a quaternionic extension of the R transform which is a generating function for planar (non-crossing) cumulants. We demonstrate that the quaternionic R transform generates all connected averages of all distinct powers of X and its hermitian conjugateWe show that the R transform for gaussian elliptic laws is given by a simple linear quaternionic map R(z + wj) = x + σ 2 µe 2iφ z + wj where (z, w) is the Cayley-Dickson pair of complex numbers forming a quaternion q = (z, w) ≡ z + wj. This map has five real parameters ex, mx, φ, σ and µ. We use the R transform to calculate the limiting eigenvalue densities of several products of gaussian random matrices.

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Limits and fluctuations of p-adic random matrix products

Peski¹

2021

Sel. Math. New Ser.

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We study the distribution of singular numbers of products of certain classes of padic random matrices, as both the matrix size and number of products go to ∞ simultaneously. In this limit, we prove convergence of the local statistics to a new random point configuration on Z, defined explicitly in terms of certain intricate mixed q-series/exponential sums. This object may be viewed as a nontrivial p-adic analogue of the interpolating distributions of Akemann-Burda-Kieburg [6], which generalize the sine and Airy kernels and govern limits of complex matrix products. Our proof uses new Macdonald process computations and holds for matrices with iid additive Haar entries, corners of Haar matrices from GLN (Zp), and the p-adic analogue of Dyson Brownian motion studied in [80]. Contents 1. Introduction 1 2. p-adic matrix background 3. Symmetric function background 4. The limit of the Plancherel/principal Hall-Littlewood measure 5. Residue expansions 6. Tightness and the limiting random variable 7. An indeterminate moment problem 8. Examples of residue formula for L k,t,χ 9. The case of pure α specializations 10. From processes of infinitely many particles to finite matrix bulk limits Appendix A. Parallels with complex matrix products Appendix B. A Hall-Littlewood proof of [80, Theorem 1.2] References

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Universal distribution of Lyapunov exponents for products of Ginibre matrices

Cited by 45 publications

References 80 publications

Equality of Lyapunov and Stability Exponents for Products of Isotropic Random Matrices

Equality of Lyapunov and Stability Exponents for Products of Isotropic Random Matrices

QuaternionicRtransform and non-Hermitian random matrices

Limits and fluctuations of p-adic random matrix products

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