Effects of ramification and connectivity degree on site percolation threshold on regular lattices and fractal networks

Balankin, Alexander S.; Martínez-Cruz, M.A.; Álvarez-Jasso, M.D.; Patiño-Ortiz, Miguel; Patiño-Ortiz, Julián

doi:10.1016/j.physleta.2018.12.018

Cited by 9 publications

(6 citation statements)

References 57 publications

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“…This procedure is recursively repeated with each of (𝑁 − 𝑀) remaining subsquares, ad infinitum. The standard Sierpiński carpets display many fascinating topological properties that appear in a vast variety of contexts [129][130][131][132][133][134][135][136][137][138][139][140][141]. It was found that the degree (averaged coordination number) of the prefractal carpets increases with the number of iterations 𝑘 as…”

Section: The Standard Sierpiński Carpetsmentioning

confidence: 99%

A Brief Survey of Paradigmatic Fractals from a Topological Perspective

Ortíz

Martínez-Cruz

et al. 2023

Fractal Fract

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The key issues in fractal geometry concern scale invariance (self-similarity or self-affinity) and the notion of a fractal dimension D which exceeds the topological dimension d. In this regard, we point out that the constitutive inequality D>d can have either a geometric or topological origin, or both. The main topological features of fractals are their connectedness, connectivity, ramification, and loopiness. We argue that these features can be specified by six basic dimension numbers which are generally independent from each other. However, for many kinds of fractals, the number of independent dimensions may be reduced due to the peculiarities of specific kinds of fractals. Accordingly, we survey the paradigmatic fractals from a topological perspective. Some challenging points are outlined.

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Section: The Standard Sierpiński Carpetsmentioning

confidence: 99%

A Brief Survey of Paradigmatic Fractals from a Topological Perspective

Ortíz

Martínez-Cruz

et al. 2023

Fractal Fract

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“…The infinite version of these fractals have Hausdorff dimensions ln(3)/ ln(2) ≈ 1.58, ln(8)/ ln(3) ≈ 1.89, and ln(5)/ ln(3) ≈ 1.46, respectively. They also differ in ramification number [25]. If we imagine all the nearest-neighbor sites being linked by a bond, an arbitrarily large piece of the T-fractal can be isolated by cutting one bond, while that number is two for the Sierpinski triangle and infinite for the Sierpinski carpet.…”

Section: The Lattice Laughlin Wavefunctionmentioning

confidence: 99%

Properties of Laughlin states on fractal lattices

Jha

Nielsen

2023

J. Stat. Mech.

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Laughlin states have recently been constructed on fractal lattices and have been shown to be topological in such systems. Some of their properties are, however, quite different from the two-dimensional case. On the Sierpinski triangle, for instance, the entanglement entropy shows oscillations as a function of particle number and does not obey the area law despite being topologically ordered, and the particle density is non-uniform in the bulk. Here, we investigate these deviant properties in greater detail on the Sierpinski triangle, and we also study the properties on the Sierpinski carpet and the T-fractal. We find that the density variations across the fractal are present for all the considered fractal lattices and for most choices of the number of particles. The size of anyons inserted into the lattice Laughlin state also varies with position on the fractal lattice. We observe that quasiholes and quasiparticles have similar sizes and that the size of the anyons typically increases with decreasing Hausdorff dimension. As opposed to periodic lattices in two dimensions, the Sierpinski triangle and carpet have inner edges. We construct trial states for both inner and outer edge states. We find that oscillations of the entropy as a function of particle number are present for the T-fractal, but not for the Sierpinski carpet. Finally, we observe deviations from the area law for several different bipartitions on the Sierpinski triangle.

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“…The scaling behavior of the fracture system connectivity is characterized by the connectivity dimension d [76]. Furthermore, the connectivity dimension along with the fractal loopiness index Λ loop determines the numbers of the effective spatial (n γ ) and dynamical (d s ) degrees of freedom of random walker in the fracture system [77][78][79]. For readers' convenience, the definitions of basic dimension numbers are summarized in the Table 2.…”

Section: Connectivitymentioning

confidence: 99%

Fractal Features of Fracture Networks and Key Attributes of Their Models

et al. 2023

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This work is devoted to the modeling of fracture networks. The main attention is focused on the fractal features of the fracture systems in geological formations and reservoirs. Two new kinds of fracture network models are introduced. The first is based on the Bernoulli percolation of straight slots in regular lattices. The second explores the site percolation in scale-free networks embedded in the two- and three-dimensional lattices. The key attributes of the model fracture networks are sketched. Surprisingly, we found that the number of effective spatial degrees of freedom of the scale-free fracture network models is determined by the network embedding dimension and does not depend on the degree distribution. The effects of degree distribution on the other fractal features of the model fracture networks are scrutinized.

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Effects of ramification and connectivity degree on site percolation threshold on regular lattices and fractal networks

Cited by 9 publications

References 57 publications

A Brief Survey of Paradigmatic Fractals from a Topological Perspective

A Brief Survey of Paradigmatic Fractals from a Topological Perspective

Properties of Laughlin states on fractal lattices

Fractal Features of Fracture Networks and Key Attributes of Their Models

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