Equality of Lyapunov and stability exponents for products of isotropic random matrices

Reddy, Nanda Kishore

doi:10.48550/arxiv.1601.02888

Cited by 3 publications

(9 citation statements)

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“…, 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. Proposition 5.1 (Central Limit Theorem for Lyapunov and Stability Exponents [55, Theorem 11]).…”

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%

“…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. We provide a sketch of an alternative proof which is based on the spherical transform and which is reminiscent of the standard Fourier analytical proof of the central limit theorem for sums of random vectors.…”

Section: Introductionmentioning

confidence: 99%

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Products of random matrices from polynomial ensembles

Kieburg¹,

Kösters²

2019

Ann. Inst. H. Poincaré Probab. Statist.

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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.

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Section: Central Limit Theorem For Lyapunov Exponentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Products of random matrices from polynomial ensembles

Kieburg¹,

Kösters²

2019

Ann. Inst. H. Poincaré Probab. Statist.

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“…This can be easiest seen in (IV. 20) where the Gaussain in j can be replaced by a Dirac delta function and the remaining integral in r is a simple exponential function leading to the sinus cardinalis in (IV.13).…”

Section: Critical Regime In the Bulkmentioning

confidence: 99%

“…Previous results were obtained either at fixed M when N → ∞, e.g., see [7][8][9][10][11][12][13][14], leading to the Meijer-G kernel at the hard edge [13] or the sine-and Airy-kernel in the bulk and at the soft edge [14], respectively. Or, the limit with N fixed with M → ∞ was considered, e.g., see [10,[15][16][17][18][19][20][21], leading to picket fence statistics [15]. A review on more recent developments is given in [22].…”

Section: Introductionmentioning

confidence: 99%

Universality of local spectral statistics of products of random matrices

Akemann,

Burda,

Kieburg

2020

Preprint

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We derive exact analytical expressions for correlation functions of singular values of the product of M Ginibre matrices of size N in the double scaling limit M, N → ∞. The singular value statistics is described by a determinantal point process with a kernel that interpolates between GUE statistic and Dirac-delta (picket-fence) statistic. In the thermodynamic limit, N → ∞, the interpolation parameter is given by the limiting quotient a = N/M . One of our goals is to find an explicit form of the kernel at the hard edge, in the bulk and at the soft edge for any a. We find that in addition to the standard scaling regimes, there is a new transitional regime which interpolates between the hard edge and the bulk. We conjecture that these results are universal, and that they apply to a broad class of products of random matrices from the Gaussian basin of attraction, including correlated matrices. We corroborate this conjecture by numerical simulations. Additionally, we show that the local spectral statistics of the considered random matrix products is identical with the local statistics of Dyson Brownian motion with the initial condition given by equidistant positions, with the crucial difference that this equivalence holds only locally. Finally, we have identified a mesoscopic spectral scale at the soft edge which is crucial for the unfolding of the spectrum.

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

Akemann¹,

Burda²,

Kieburg³

2019

EPL

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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 ×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 also the case for many chaotic quantum systems, we identify a critical double scaling limit N ∼ M for the rest of the spectrum. It interpolates between the known deterministic behaviour of the Lyapunov exponents for M N (or N fixed) and universal random matrix statistics for M N (or M fixed), characterising chaotic behaviour. After unfolding this agrees with Dyson's Brownian Motion starting from equidistant positions in the bulk and at the soft edge of the spectrum. This universality statement is further corroborated by numerical experiments, multiplying different kinds of random matrices. It leads us to conjecture a much wider applicability in complex systems, that display a transition from deterministic to chaotic behaviour.

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Equality of Lyapunov and stability exponents for products of isotropic random matrices

Cited by 3 publications

References 0 publications

Products of random matrices from polynomial ensembles

Products of random matrices from polynomial ensembles

Universality of local spectral statistics of products of random matrices

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

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