On Limit Theorems for the Product of Random Matrices

Tutubalin, V. N.

doi:10.1137/1110002

Cited by 93 publications

(48 citation statements)

References 3 publications

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“…84 This theorem states that the Lyapunov exponents of products of a finite number of random matrices are random numbers whose distribution approaches Gaussian for large sample lengths.…”

Section: Construction and Simulation Of Random Networkmentioning

confidence: 99%

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Geometrically disordered network models, quenched quantum gravity, and critical behavior at quantum Hall plateau transitions

et al. 2017

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Recent results for the critical exponent of the localization length at the integer quantum Hall transition differ considerably between experimental (νexp ≈ 2.38) and numerical (νCC ≈ 2.6) values obtained in simulations of the Chalker-Coddington (CC) network model. The difference is at least partially due to effects of the electron-electron interaction present in experiments. Here we propose a mechanism that changes the value of ν even within the single-particle picture. We revisit the arguments leading to the CC model and consider more general networks with structural disorder. Numerical simulations of the new model lead to the value ν ≈ 2.37. We argue that in a continuum limit the structurally disordered model maps to free Dirac fermions coupled to various random potentials (similar to the CC model) but also to quenched two-dimensional quantum gravity. This explains the possible reason for the considerable difference between critical exponents for the CC model and the structurally disordered model. We extend our results to network models in other symmetry classes.

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“…84 This theorem states that the Lyapunov exponents of products of a finite number of random matrices are random numbers whose distribution approaches Gaussian for large sample lengths.…”

Section: Construction and Simulation Of Random Networkmentioning

confidence: 99%

“…We calculated 624 disorder realizations for any combination of M = 20, 40, 60, 80, 100, 120, 140, 160, 180, 200 and x = 0.08/12 · [0, 1, 2, 3,4,5,6,7,8,9,10,11,12] for fixed L = 5 000 000. Our goal is to check whether the central limit theorem 84 also works in the case of randomness of the network or not. Fig.…”

Section: Weights and Errorsmentioning

confidence: 99%

Geometrically disordered network models, quenched quantum gravity, and critical behavior at quantum Hall plateau transitions

et al. 2017

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“…These results involved stationary measures and eventually led to a general analysis of noncommutative "large numbers" phenomena on semisimple Lie groups. Some representative works are Furstenberg [12,13], Grenander [17], Guivarc'h [18], Guivarc'h et al [19] and Tutubalin [27], and the bibliographies there.…”

Section: Introductionmentioning

confidence: 99%

Central limit theorem for products of random matrices

Berger¹

1984

Trans. Amer. Math. Soc.

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ABSTRACT. Using the semigroup product formula of P. Chernoff, a central limit theorem is derived for products of random matrices. Applications are presented for representations of solutions to linear systems of stochastic differential equations, and to the corresponding partial differential evolution equations. Included is a discussion of stochastic semigroups, and a stochastic version of the Lie-Trotter product formula.

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“…Rather than calculating the LEs for very long chains, one can obtain them for a large number N r of shorter chains of length L. Then, by calculating the ensemble average LE " and its standard deviation and making use of the central limit theorem [18], the error of " can be estimated…”

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confidence: 99%

Numerical Study of the Localization Length Critical Index in a Network Model of Plateau-Plateau Transitions in the Quantum Hall Effect

Amado

Sedrakyan

et al. 2011

Phys. Rev. Lett.

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We calculate numerically the localization length critical index within the Chalker-Coddington model of the plateau-plateau transitions in the quantum Hall effect. We report a finite-size scaling analysis using both the traditional power-law corrections to the scaling function and the inverse logarithmic ones, which provided a more stable fit resulting in the localization length critical index ¼ 2:616 AE 0:014. We observe an increase of the critical exponent with the system size, which is possibly the origin of discrepancies with early results obtained for smaller systems. DOI: 10.1103/PhysRevLett.107.066402 PACS numbers: 71.30.+h, 71.23.An, 72.15.Rn Plateau-plateau transitions in the quantum Hall effect have been one of the most challenging problems in condensed matter physics during the past two decades. It is an interesting example of the localization-delocalization transition in two-dimensional disordered systems, where a quantum critical point appears due to the breaking of time reversal symmetry. One of the important problems in this area of research is the formulation of a quantum field theory describing the transition. The first suggestion in this respect appeared in Ref. [1], where the authors noticed that the presence of the topological term in the nonlinear sigma model formulation of the problem can result in the occurrence of delocalized states under strong magnetic fields.Later, Chalker and Coddington [2] formulated a phenomenological model of quantum percolation based on a transfer-matrix approach (referred to as the CC model hereafter). The numerical value 2:5 AE 0:5 of the critical index of the Lyapunov exponent (LE) calculated within the CC model (see Ref.[3] for a review) was in good agreement with the experimentally measured localization length index ¼ 2:4 in the quantum Hall effect [4]. This success motivated considerable interest in the CC model and stimulated its further investigation until the present day [5][6][7][8][9][10][11][12][13][14]. In early studies the continuum limit of the CC model was related to replicas of ordinary spin chains [6], a Hubbard-like model [7], and supersymmetric spin chains [8,9]. In Refs. [10,11], the continuum limit was also related to the conformal field theory of the Wess-Zumino-Witten-Novikov (WZWN) type. Analyzing the representations of the PSLð2j2Þ conformal field theory, they found one which gives a reasonable value of 16=7 ' 2:286 for the localization length index. Moreover, multifractal scaling indices of the CC model were predicted to depend quadratically on the parameter q of the multifractal analysis within the WZWN model.Most intriguing developments in the plateau-plateau transition problem were reported later in Refs. [15,16], where the multifractal behavior of the CC model was investigated. In both papers, approximately quartic deviation from the exact quadratic dependence of the multifractal indices on the parameter q, predicted in Refs. [10,11], was observed. The latter suggested that the validity of the supersymmetric WZWN approach to plate...

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On Limit Theorems for the Product of Random Matrices

Cited by 93 publications

References 3 publications

Geometrically disordered network models, quenched quantum gravity, and critical behavior at quantum Hall plateau transitions

Geometrically disordered network models, quenched quantum gravity, and critical behavior at quantum Hall plateau transitions

Central limit theorem for products of random matrices

Numerical Study of the Localization Length Critical Index in a Network Model of Plateau-Plateau Transitions in the Quantum Hall Effect

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