Ubiquitous Fractal Dimension of Optimal Paths

Andrade, José S.; Reis, Saulo D. S.; Oliveira, Erneson A.; Fehr, E.; Herrmann, Hans J.

doi:10.1109/mcse.2011.16

Cited by 16 publications

(22 citation statements)

References 24 publications

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“…8). On the square lattice, the fractal dimension of the exter-nal perimeter was shown to be related to several other models [16,27]. The value reported here for the simplecubic lattice agrees within its error bars with the one for watersheds and the optimal path cracking [28] as well as with the set of bridges in bridge percolation [19].…”

Section: The Gaussian Model On the Simple-cubic Latticesupporting

confidence: 70%

Gaussian model of explosive percolation in three and higher dimensions

2011

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The Gaussian model of discontinuous percolation, recently introduced by Araújo and Herrmann [Phys. Rev. Lett., 105, 035701 (2010)], is numerically investigated in three dimensions, disclosing a discontinuous transition. For the simple-cubic lattice, in the thermodynamic limit, we report a finite jump of the order parameter, J = 0.415 ± 0.005. The largest cluster at the threshold is compact, but its external perimeter is fractal with fractal dimension dA = 2.5 ± 0.2. The study is extended to hypercubic lattices up to six dimensions and to the mean-field limit (infinite dimension). We find that, in all considered dimensions, the percolation transition is discontinuous. The value of the jump in the order parameter, the maximum of the second moment, and the percolation threshold are analyzed, revealing interesting features of the transition and corroborating its discontinuous nature in all considered dimensions. We also show that the fractal dimension of the external perimeter, for any dimension, is consistent with the one from bridge percolation and establish a lower bound for the percolation threshold of discontinuous models with finite number of clusters at the threshold.

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Section: The Gaussian Model On the Simple-cubic Latticesupporting

confidence: 70%

Gaussian model of explosive percolation in three and higher dimensions

2011

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“…It is also known that the minimum path dimension takes the value d mst 1.22 0.01 m =  ( ) [4]. This is also the minimum path dimension of strands in invasion percolation, optimal path cracks and fractal watershed lines [5]. From equation (6) we see that the connectivity dimension for MSTs should be roughly d mst 2 1.22 1.64…”

Section: Mst Universality Classmentioning

confidence: 92%

“…The averaged mass B s, ℓ | ( )| measured using Euclidean length is nothing more that the connected mass in equation (5). Hence, we expect that…”

Section: Non-euclidean Fractal Measuresmentioning

confidence: 99%

“…The MST universality class is a famous one, to which many systems have been argued to belong [5]. Minimal paths on MST, minimal paths on invasion percolation clusters and watershed lines are examples of random planar curves with the same apparent fractal dimension of 1.22 [5]. Although the frictional finger trees and the MSTs follow similar dynamical construction rules it is not obvious that they share universality class.…”

Section: Introductionmentioning

confidence: 99%

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Geometric universality and anomalous diffusion in frictional fingers

et al. 2019

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Frictional finger trees are patterns emerging from non-equilibrium processes in particle-fluid systems. Their formation share several properties with growth algorithms for minimum spanning trees (MSTs) in random energy landscapes. We propose that the frictional finger trees are indeed in the same geometric universality class as the MSTs, which is checked using updated numerical simulation algorithms for frictional fingers. We also propose a theoretical model for anomalous diffusion in these patterns, and discuss the role of diffusion as a tool to classify geometry.

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“…During this process, transport in the system remains possible, with increasing cost, until the formation of a path of destroyed sites which disconnects the system into two parts. This process, denoted as optimal path cracking, was introduced recently by Andrade et al [13], who discovered that the shortest path of destroyed sites necessary to disconnect the system has a fractal dimension of 1.22 ± 0.02, an interesting exponent also found in several other systems [14,15,30,32,33,35].…”

Section: A the Optimal Pathmentioning

confidence: 99%

Optimal-path cracks in correlated and uncorrelated lattices

et al. 2011

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The optimal path crack model on uncorrelated surfaces, recently introduced by Andrade et al. (Phys. Rev. Lett. 103, 225503, 2009), is studied in detail and its main percolation exponents computed. In addition to β/ν = 0.46 ± 0.03 we report, for the first time, γ/ν = 1.3 ± 0.2 and τ = 2.3 ± 0.2. The analysis is extended to surfaces with spatial long-range power-law correlations, where non-universal fractal dimensions are obtained when the degree of correlation is varied. The model is also considered on a three-dimensional lattice, where the main crack is found to be a surface with a fractal dimension of 2.46 ± 0.05.

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Ubiquitous Fractal Dimension of Optimal Paths

Cited by 16 publications

References 24 publications

Gaussian model of explosive percolation in three and higher dimensions

Gaussian model of explosive percolation in three and higher dimensions

Geometric universality and anomalous diffusion in frictional fingers

Optimal-path cracks in correlated and uncorrelated lattices

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