First and second moments of the size distribution of bond percolation clusters on rings, paths and stars

2020 ◽  
Vol 161 ◽  
pp. 108714
Author(s):  
Petar Jevtić ◽  
Nicolas Lanchier ◽  
Axel La Salle
Keyword(s):  
Size Distribution ◽  
Bond Percolation ◽  
Percolation Clusters ◽  
Second Moments

scholarly journals Almost Giant Clusters for Percolation on Large Trees with Logarithmic Heights

2013 ◽  
Vol 50 (3) ◽  
pp. 603-611 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Random Trees ◽  
Applied Probability ◽  
Bond Percolation ◽  
General Criterion ◽  
Asymptotic Results ◽  
Percolation Clusters ◽  
Large Trees ◽  
Logarithmic Height

This paper is based on works presented at the 2012 Applied Probability Trust Lecture in Sheffield; its main purpose is to survey the recent asymptotic results of Bertoin (2012a) and Bertoin and Uribe Bravo (2012b) about Bernoulli bond percolation on certain large random trees with logarithmic height. We also provide a general criterion for the existence of giant percolation clusters in large trees, which answers a question raised by David Croydon.

Hydrodynamic radii of diffusion‐limited aggregates and bond‐percolation clusters

1988 ◽  
Vol 89 (9) ◽  
pp. 5887-5889 ◽  
Author(s):  
Zhong‐Ying Chen ◽  
Paul C. Weakliem ◽  
Paul Meakin
Keyword(s):  
Bond Percolation ◽  
Percolation Clusters ◽  
Diffusion Limited

scholarly journals A Boltzmann Approach to Percolation on Random Triangulations

2019 ◽  
Vol 71 (1) ◽  
pp. 1-43 ◽  
Author(s):  
Olivier Bernardi ◽  
Nicolas Curien ◽  
Grégory Miermont
Keyword(s):  
Percolation Model ◽  
Bond Percolation ◽  
Critical Percolation ◽  
Site Percolation ◽  
Percolation Clusters ◽  
Percolation Parameter ◽  
Stable Map ◽  
Percolation Thresholds ◽  
Boltzmann Model ◽  
Independent Derivation

AbstractWe study the percolation model on Boltzmann triangulations using a generating function approach. More precisely, we consider a Boltzmann model on the set of finite planar triangulations, together with a percolation configuration (either site-percolation or bond-percolation) on this triangulation. By enumerating triangulations with boundaries according to both the boundary length and the number of vertices/edges on the boundary, we are able to identify a phase transition for the geometry of the origin cluster. For instance, we show that the probability that a percolation interface has length$n$decays exponentially with$n$except at a particular value$p_{c}$of the percolation parameter$p$for which the decay is polynomial (of order$n^{-10/3}$). Moreover, the probability that the origin cluster has size$n$decays exponentially if$p<p_{c}$and polynomially if$p\geqslant p_{c}$.The critical percolation value is$p_{c}=1/2$for site percolation, and$p_{c}=(2\sqrt{3}-1)/11$for bond percolation. These values coincide with critical percolation thresholds for infinite triangulations identified by Angel for site-percolation, and by Angel and Curien for bond-percolation, and we give an independent derivation of these percolation thresholds.Lastly, we revisit the criticality conditions for random Boltzmann maps, and argue that at$p_{c}$, the percolation clusters conditioned to have size$n$should converge toward the stable map of parameter$\frac{7}{6}$introduced by Le Gall and Miermont. This enables us to derive heuristically some new critical exponents.

Different approximate models of the sectional method for nanoparticle Brownian coagulation

2015 ◽  
Vol 25 (2) ◽  
pp. 438-448 ◽  
Author(s):  
Zhongli Chen ◽  
Fangyang Yuan ◽  
R.J. Jiang
Keyword(s):  
Size Distribution ◽  
Time Evolution ◽  
Size Range ◽  
Approximate Model ◽  
Brownian Coagulation ◽  
Content Type ◽  
Approximation Factor ◽  
Advantages And Disadvantages ◽  
Second Moments ◽  
Approximate Models

Purpose – The original v2-based sectional method assumes that the selected property quantity of particles is uniformly distributed in each section, which makes particle size distribution (PSD) fluctuate dramatically in the entire size range. The number concentration in each section as well as the zeroth moment of PSD also cannot be correctly predicted in case there are not enough sections used in calculation. In order to provide a more appropriate representation of PSD, different approximate models are used to close the conservation equations. The paper aims to discuss these issues. Design/methodology/approach – The uniform distribution of the selected property quantity of particles in each section is not necessarily satisfied. Instead, the distribution is approximated using an expression with an approximation factor. Different models are investigated on recovering the initial size distribution and predicting the time evolution of size distribution as well as the first three moments so that the advantages and disadvantages of each model can be compared. Findings – The approximate model with an approximation factor of 0.8 is capable of predicting the time evolution of the zeroth moment accurately no matter how many sections are used in simulations. The original v2-based model is recommended to calculate the first and second moments as long as the section number is larger than 50, otherwise, the model with an approximation factor of 0.15 would be a preferred choice. Originality/value – Different approximate models can be used to improve the accuracy of the results supposing we know which moment is of great importance in calculation.

scholarly journals Almost Giant Clusters for Percolation on Large Trees with Logarithmic Heights

2013 ◽  
Vol 50 (03) ◽  
pp. 603-611 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Random Trees ◽  
Applied Probability ◽  
Bond Percolation ◽  
General Criterion ◽  
Asymptotic Results ◽  
Percolation Clusters ◽  
Large Trees ◽  
Logarithmic Height

This paper is based on works presented at the 2012 Applied Probability Trust Lecture in Sheffield; its main purpose is to survey the recent asymptotic results of Bertoin (2012a) and Bertoin and Uribe Bravo (2012b) about Bernoulli bond percolation on certain large random trees with logarithmic height. We also provide a general criterion for the existence of giant percolation clusters in large trees, which answers a question raised by David Croydon.

A NEW TYPE OF DYNAMICAL EXCITATIONS FOR PERCOLATION CLUSTERS

Fractals ◽  
1993 ◽  
Vol 01 (04) ◽  
pp. 959-962 ◽  
Author(s):  
M. KOLB
Keyword(s):  
Numerical Simulations ◽  
Three Dimensions ◽  
Bond Percolation ◽  
Scalar Model ◽  
Local Connectivity ◽  
Scaling Properties ◽  
Percolation Clusters ◽  
Chemical Distance ◽  
Dimension Change ◽  
New Type

By increasing the local connectivity of percolation clusters, structures with different scaling properties are generated. Both the topology, characterized by the chemical distance, and the dynamics, measured by the spectral dimension, change. For bond percolation in two (three) dimensions, new exponents have been determined by means of numerical simulations: dmin=1.02(4) (1.09(6)) and, for the scalar model, ds=1.56(7) (1–76(8)).

Second-Order Light Scattering and Local Anisotropy of Diffusion-Limited Aggregates and Bond-Percolation Clusters

1987 ◽  
Vol 58 (19) ◽  
pp. 1996-1999 ◽  
Author(s):  
Zhong-Ying Chen ◽  
Paul Weakliem ◽  
William M. Gelbart ◽  
Paul Meakin
Keyword(s):  
Light Scattering ◽  
Second Order ◽  
Bond Percolation ◽  
Local Anisotropy ◽  
Percolation Clusters ◽  
Diffusion Limited
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