Isotropy properties of the multi-step Markov symbolic sequences

Apostolov, S. S.; Mayzelis, Z. A.; Usatenko, O. V.; Yampol’skiı̆, V. A.

doi:10.1016/j.physa.2006.10.069

Cited by 4 publications

(4 citation statements)

References 24 publications

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“…Since finding the correlation function for such chains brings about considerable technical difficulties, we examine the required statistical properties of the Ising chain directly from equation (4). 15), and lowtemperature, equation (22), limits, respectively. Right panel (T = 0.6): solid curve corresponds to the low-temperature, equation (22), limit.…”

Section: Magnetic Properties Of the Ising Chain In The Low-temperatur...mentioning

confidence: 99%

“…15), and lowtemperature, equation (22), limits, respectively. Right panel (T = 0.6): solid curve corresponds to the low-temperature, equation (22), limit. The RMS magnetization in the absence of the magnetic field versus length L of a segment is presented in the insets.…”

Section: Magnetic Properties Of the Ising Chain In The Low-temperatur...mentioning

confidence: 99%

“…, s i−1 ). The statistical properties of such chains were studied in detail in [7,[19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

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Non-additive properties of finite 1D Ising chains with long-range interactions

Apostolov¹,

Mayzelis²,

Usatenko³

et al. 2009

J. Phys. A: Math. Theor.

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We study the statistical properties of Ising spin chains with finite (although arbitrary large) range of interaction between the elements. We examine mesoscopic subsystems (fragments of an Ising chain) with the lengths comparable with the interaction range. The equivalence of the Ising chains and the multi-step Markov sequences is used for calculating different non-additive statistical quantities of a chain and its fragments. In particular, we study the variance of fluctuating magnetization of fragments, magnetization of the chain in the external magnetic field, etc.Asymptotical expressions for the non-additive energy and entropy of the mesoscopic fragments are derived in the limiting cases of weak and strong interactions.

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Section: Magnetic Properties Of the Ising Chain In The Low-temperatur...mentioning

confidence: 99%

Section: Magnetic Properties Of the Ising Chain In The Low-temperatur...mentioning

confidence: 99%

See 1 more Smart Citation

Non-additive properties of finite 1D Ising chains with long-range interactions

Apostolov¹,

Mayzelis²,

Usatenko³

et al. 2009

J. Phys. A: Math. Theor.

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“…The sequences produced by even filtering functions are isotropic. Isotropy properties of the multi-step Markov dichotomous sequences were earlier studied in [35].…”

Section: Weak Short-range Correlationsmentioning

confidence: 99%

Bunches of random cross-correlated sequences

Maystrenko¹,

Melnik²,

Pritula³

et al. 2013

J. Phys. A: Math. Theor.

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The statistical properties of random cross-correlated sequences constructed by the convolution method (likewise referred to as the Rice or the inverse Fourier transformation) are examined. We clarify the meaning of the filtering function-the kernel of the convolution operator-and show that it is the value of the cross-correlation function which describes correlations between the initial white noise and constructed correlated sequences. The matrix generalization of this method for constructing a bunch of N cross-correlated sequences is presented. Algorithms for their generation are reduced to solving the problem of decomposition of the Fourier transform of the correlation matrix into a product of two mutually conjugate matrices. Different decompositions are considered. The limits of weak and strong correlations for the one-point probability and pair correlation functions of sequences generated by the method under consideration are studied. Special cases of heavy-tailed distributions of the generated sequences are analyzed. We show that, if the filtering function is rather smooth, the distribution function of generated variables has the Gaussian or Lévy form depending on the analytical properties of the distribution (or characteristic) functions of the initial white noise. Anisotropic properties of statistically homogeneous random sequences related to the asymmetry of a filtering function are revealed and studied. These asymmetry properties are expressed in terms of the third-or fourth-order correlation functions. Several examples of the construction of correlated chains with a predefined correlation matrix are given.

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