Aspects of the Fractal Percolation Process

1995 ◽  
pp. 113-143 ◽  
Author(s):  
Lincoln Chayes
Keyword(s):  
Percolation Process ◽  
Fractal Percolation

Random fractals and their intersection with winning sets

2021 ◽  
pp. 1-30
Author(s):  
YIFTACH DAYAN
Keyword(s):  
Hausdorff Dimension ◽  
General Result ◽  
Countable Collection ◽  
Percolation Process ◽  
Fractal Percolation ◽  
Random Fractals ◽  
Affine Hyperplane ◽  
Badly Approximable

Abstract We show that fractal percolation sets in $\mathbb{R}^{d}$ almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if $E\subset\mathbb{R}^{d}$ is a realisation of a fractal percolation process, then almost surely (conditioned on $E\neq\emptyset$ ), for every countable collection $\left(f_{i}\right)_{i\in\mathbb{N}}$ of $C^{1}$ diffeomorphisms of $\mathbb{R}^{d}$ , $\dim_{H}\left(E\cap\left(\bigcap_{i\in\mathbb{N}}f_{i}\left(\text{BA}_{d}\right)\right)\right)=\dim_{H}\left(E\right)$ , where $\text{BA}_{d}$ is the set of badly approximable vectors in $\mathbb{R}^{d}$ . We show this by proving that E almost surely contains hyperplane diffuse subsets which are Ahlfors-regular with dimensions arbitrarily close to $\dim_{H}\left(E\right)$ . We achieve this by analysing Galton–Watson trees and showing that they almost surely contain appropriate subtrees whose projections to $\mathbb{R}^{d}$ yield the aforementioned subsets of E. This method allows us to obtain a more general result by projecting the Galton–Watson trees against any similarity IFS whose attractor is not contained in a single affine hyperplane. Thus our general result relates to a broader class of random fractals than fractal percolation.

scholarly journals Geometric functionals of fractal percolation

2020 ◽  
Vol 52 (4) ◽  
pp. 1085-1126
Author(s):  
Michael A. Klatt ◽  
Steffen Winter
Keyword(s):  
Topological Phase ◽  
Explicit Formulas ◽  
Percolation Process ◽  
Intrinsic Volumes ◽  
Fractal Percolation ◽  
Fractal Models ◽  
Topological Phase Transition ◽  
Finite Approximations ◽  
Percolation Thresholds ◽  
Transition Points

AbstractFractal percolation exhibits a dramatic topological phase transition, changing abruptly from a dust-like set to a system-spanning cluster. The transition points are unknown and difficult to estimate. In many classical percolation models the percolation thresholds have been approximated well using additive geometric functionals, known as intrinsic volumes. Motivated by the question of whether a similar approach is possible for fractal models, we introduce corresponding geometric functionals for the fractal percolation process F. They arise as limits of expected functionals of finite approximations of F. We establish the existence of these limit functionals and obtain explicit formulas for them as well as for their finite approximations.

scholarly journals Effect of Hydrophobic Silica Nanochannel Structure on the Running Speed of a Colloidal Damper

2021 ◽  
Vol 11 (15) ◽  
pp. 6808
Author(s):  
Gengbiao Chen ◽  
Zhiwen Liu
Keyword(s):  
Fractal Dimension ◽  
Silica Gel ◽  
Fractal Theory ◽  
Percolation Model ◽  
Silica Gels ◽  
Practical Significance ◽  
Running Speed ◽  
Fractal Percolation ◽  
Micropore Structure ◽  
Hydrophobic Silica

A colloidal damper (CD) can dissipate a significant amount of vibrations and impact energy owing to the interface power that is generated when it is used. It is of great practical significance to study the influence of the nanochannel structure of hydrophobic silica gel in the CD damping medium on the running speed of the CD. The fractal theory was applied to observe the characteristics of the micropore structure of the hydrophobic silica gel by scanning electron microscopy (SEM), the primary particles were selected to carry out fractal analysis, and the two-dimensional fractal dimension of the pore area and the tortuous fractal dimension of the hydrophobic silica gel pore structure were calculated. The fractal percolation model of water in hydrophobic silica nanochannels based on the slip theory could thus be obtained. This model revealed the relationship between the micropore structure parameters of the silica gel and the running speed of the CD. The CD running speed increases with the addition of grafted molecules and the reduction in pore size of the silica gel particles. Continuous loading velocity testing of the CD loaded with hydrophobic silica gels with different pore structures was conducted. By comparing the experimental results with the calculation results of the fractal percolation model, it was determined that the fractal percolation model can better characterize the change trend of the CD running velocity for the first loading, but the fractal dimension was changed from the second loading, caused by the small amount of water retained in the nanochannel, leading to the failure of fractal characterization.

scholarly journals Non-existence of bi-infinite geodesics in the exponential corner growth model

2020 ◽  
Vol 8 ◽  
Author(s):  
Márton Balázs ◽  
Ofer Busani ◽  
Timo Seppäläinen
Keyword(s):  
Growth Model ◽  
Coarse Graining ◽  
Exponential Weights ◽  
Percolation Process ◽  
Last Passage Percolation ◽  
Corner Growth Model ◽  
And Control

Abstract This paper gives a self-contained proof of the non-existence of nontrivial bi-infinite geodesics in directed planar last-passage percolation with exponential weights. The techniques used are couplings, coarse graining, and control of geodesics through planarity and estimates derived from increment-stationary versions of the last-passage percolation process.

Percolation phase transition of static and growing networks under a weighted function

2016 ◽  
Vol 27 (07) ◽  
pp. 1650082 ◽  
Author(s):  
Xiao Jia ◽  
Jin-Song Hong ◽  
Ya-Chun Gao ◽  
Hong-Chun Yang ◽  
Chun Yang ◽  
...  
Keyword(s):  
Phase Transition ◽  
Continuous Phase ◽  
Weighted Function ◽  
Percolation Process ◽  
Practical Applications ◽  
Network Intervention ◽  
Growing Networks ◽  
Discontinuous Phase ◽  
And Control ◽  
Growing Network

We investigate the percolation phase transitions in both the static and growing networks where the nodes are sampled according to a weighted function with a tunable parameter [Formula: see text]. For the static network, i.e. the number of nodes is constant during the percolation process, the percolation phase transition can evolve from continuous to discontinuous as the value of [Formula: see text] is tuned. Based on the properties of the weighted function, three typical values of [Formula: see text] are analyzed. The model becomes the classical Erdös–Rényi (ER) network model at [Formula: see text]. When [Formula: see text], it is shown that the percolation process generates a weakly discontinuous phase transition where the order parameter exhibits an extremely abrupt transition with a significant jump in large but finite system. For [Formula: see text], the cluster size distribution at the lower pseudo-transition point does not obey the power-law behavior, indicating a strongly discontinuous phase transition. In the case of growing network, in which the collection of nodes is increasing, a smoother continuous phase transition emerges at [Formula: see text], in contrast to the weakly discontinuous phase transition of the static network. At [Formula: see text], on the other hand, probability modulation effect shows that the nature of strongly discontinuous phase transition remains the same with the static network despite the node arrival even in the thermodynamic limit. These percolation properties of the growing networks could provide useful reference for network intervention and control in practical applications in consideration of the increasing size of most actual networks.

scholarly journals Identifying Dominant Runoff Processes at a Regional Scale – A GIS - Based Approach

2017 ◽  
Vol 11 (2) ◽  
pp. 19-33
Author(s):  
Fagbohun Babatunde Joseph ◽  
Olabode Oluwaseun Franklin ◽  
Adebola Abiodun Olufemi
Keyword(s):  
Overland Flow ◽  
Expert Knowledge ◽  
Regional Scale ◽  
Field Investigation ◽  
Deep Percolation ◽  
Flood Prediction ◽  
Percolation Process ◽  
Least Area ◽  
Expected Increase

Abstract Identifying landscapes with similar hydrological characteristics is useful for the determination of dominant runoff process (DRP) and flood prediction. Several approaches used for DRP-mapping differ in respect to time and data requirement. Manual approaches based on field investigation and expert knowledge are time consuming and difficult to implement at regional scale. Automatic GIS-based approach on the other hand require simplification of data but are easier to implement and it is applicable on regional scale. In this study, GIS-based automated approach was used to identify the DRPs in Anambra area. The result showed that Hortonian Overland Flow (HOF) has the highest coverage of 1508.3 Km2 (33.5%) followed by Deep Percolation (DP) with coverage of 1455.3 Km2 (32.3%). Subsurface Flow (SSF) is the third dominant runoff process covering 920.6 Km2 (20.4%) while Saturated Overland Flow (SOF) covers the least area of 618.4 Km2 (13.7%) of the study area. The result reveal that considerable amount of precipitated water would be infiltrated into the subsurface through deep percolation process contributing to groundwater recharge in the study area. However, it is envisaged that HOF and SOF will continue to increase due to the continuous expansion of built-up area. With the expected increase in HOF and SOF and the change in rainfall pattern associated with perpetual problem of climate change, it is paramount that groundwater conservation practices be considered to ensure continued sustainable utilization of groundwater in the study area.

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