Existence of phase transition of percolation on Sierpiński carpet lattices

Shinoda, Masato

doi:10.1239/jap/1019737983

Cited by 17 publications

(25 citation statements)

References 9 publications

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“…Shinoda [2] showed that the phase transitions can be obtained for the percolation on the Sierpinski carpet lattice, which is a self-similar graph being like the framework supporting the whole Sierpinski carpet [14,15]. Since the financial time series exhibits various behaviors of the multifractality [16,17], in the present work, the percolation system on the fractal Sierpinski carpet lattice is utilized to establish a financial price model and describe the fluctuation behavior of returns of the proposed model for different parameter values.…”

Section: Introductionmentioning

confidence: 92%

“…(2) n be the union ofS (2) n and its reflections in every coordinate hyperplane. Then we define the Sierpinski carpet as S (2) = ∞ n=0 S (2) n .…”

Section: Financial Time Series Modelmentioning

confidence: 99%

“…Then we define the Sierpinski carpet as S (2) = ∞ n=0 S (2) n . Similarly to the bond percolation mentioned above, we define the corresponding edges set of S (2) as E(S (2) ). Next we consider percolation on the lattice L(S (2) ) = (S (2) , E(S (2) )), see Fig.…”

Section: Financial Time Series Modelmentioning

confidence: 99%

“…Recent empirical research has been performed to apply statistical physics systems to financial market dynamics [1][2][3][4][5][6], since econometric modeling is vital in finance and financial time series analysis. The modeling of dynamics of forward prices is becoming a key problem in the risk management, physical assets valuation, derivatives pricing, etc.…”

Section: Introductionmentioning

confidence: 99%

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Volatility Behaviors of Financial Time Series by Percolation System on Sierpinski Carpet Lattice

Pei

Wang

2015

Fluct. Noise Lett.

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The financial time series is simulated and investigated by the percolation system on the Sierpinski carpet lattice, where percolation is usually employed to describe the behavior of connected clusters in a random graph, and the Sierpinski carpet lattice is a graph which corresponds the fractal -Sierpinski carpet. To study the fluctuation behavior of returns for the financial model and the Shanghai Composite Index, we establish a daily volatility measure -multifractal volatility (MFV) measure to obtain MFV series, which have long-range cross-correlations with squared daily return series. The autoregressive fractionally integrated moving average (ARFIMA) model is used to analyze the MFV series, which performs better when compared to other volatility series. By a comparative study of the multifractality and volatility analysis of the data, the simulation data of the proposed model exhibits very similar behaviors to those of the real stock index, which indicates somewhat rationality of the model to the market application.

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Section: Introductionmentioning

confidence: 92%

“…(2) n be the union ofS (2) n and its reflections in every coordinate hyperplane. Then we define the Sierpinski carpet as S (2) = ∞ n=0 S (2) n .…”

Section: Financial Time Series Modelmentioning

confidence: 99%

Section: Financial Time Series Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Volatility Behaviors of Financial Time Series by Percolation System on Sierpinski Carpet Lattice

Pei

Wang

2015

Fluct. Noise Lett.

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“…Lü [11] gave an alternative proof of p c < 1 using a Peierls argument. Shinoda [15] gave sufficient conditions and necessary conditions to have p c < 1 for generalized Sierpinski carpet lattices. Murai [13] studied an asymptotic behavior as d → ∞ of the critical probability of d-dimensional Sierpinski carpet lattices.…”

Section: Introductionmentioning

confidence: 99%

Non-existence of phase transition of oriented percolation on Sierpinski carpet lattices

Shinoda¹

2003

Probab Theory Relat Fields

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Near-critical 2D percolation with heavy-tailed impurities, forest fires and frozen percolation

Berg

Nolin

2021

Probab. Theory Relat. Fields

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We introduce a new percolation model on planar lattices. First, impurities (“holes”) are removed independently from the lattice. On the remaining part, we then consider site percolation with some parameter p close to the critical value $$p_c$$ p c . The mentioned impurities are not only microscopic, but allowed to be mesoscopic (“heavy-tailed”, in some sense). For technical reasons (the proofs of our results use quite precise bounds on critical exponents in Bernoulli percolation), our study focuses on the triangular lattice. We determine explicitly the range of parameters in the distribution of impurities for which the connectivity properties of percolation remain of the same order as without impurities, for distances below a certain characteristic length. This generalizes a celebrated result by Kesten for classical near-critical percolation (which can be viewed as critical percolation with single-site impurities). New challenges arise from the potentially large impurities. This generalization, which is also of independent interest, turns out to be crucial to study models of forest fires (or epidemics). In these models, all vertices are initially vacant, and then become occupied at rate 1. If an occupied vertex is hit by lightning, which occurs at a very small rate $$\zeta $$ ζ , its entire occupied cluster burns immediately, so that all its vertices become vacant. Our results for percolation with impurities are instrumental in analyzing the behavior of these forest fire models near and beyond the critical time (i.e. the time after which, in a forest without fires, an infinite cluster of trees emerges). In particular, we prove (so far, for the case when burnt trees do not recover) the existence of a sequence of “exceptional scales” (functions of $$\zeta $$ ζ ). For forests on boxes with such side lengths, the impact of fires does not vanish in the limit as $$\zeta \searrow 0$$ ζ ↘ 0 . This surprising behavior, related to the non-monotonicity of these processes, was not predicted in the physics literature.

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Existence of phase transition of percolation on Sierpiński carpet lattices

Cited by 17 publications

References 9 publications

Volatility Behaviors of Financial Time Series by Percolation System on Sierpinski Carpet Lattice

Volatility Behaviors of Financial Time Series by Percolation System on Sierpinski Carpet Lattice

Non-existence of phase transition of oriented percolation on Sierpinski carpet lattices

Near-critical 2D percolation with heavy-tailed impurities, forest fires and frozen percolation

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