From discrete to continuous percolation in dimensions 3 to 7

@article{Koza2016FromDT,
  title={From discrete to continuous percolation in dimensions 3 to 7},
  author={Zbigniew Koza and Jakub Pola},
  journal={Journal of Statistical Mechanics: Theory and Experiment},
  year={2016},
  volume={2016}
}
  • Z. KozaJ. Pola
  • Published 26 June 2016
  • Mathematics, Computer Science
  • Journal of Statistical Mechanics: Theory and Experiment
We propose a method of studying the continuous percolation of aligned objects as a limit of a corresponding discrete model. We show that the convergence of a discrete model to its continuous limit is controlled by a power-law dependency with a universal exponent θ=3/2. This allows us to estimate the continuous percolation thresholds in a model of aligned hypercubes in dimensions d=3,…,7 with accuracy far better than that attained using any other method before. We also report improved values of… 
View on IOP Publishing
arxiv.org
25 Citations
Highly Influential Citations
2
Background Citations
6
Results Citations
1

25 Citations

Percolation of hyperspheres in dimensions 3 to 5: from discrete to continuous

We study the onset of percolation of overlapping discrete hyperspheres on hypercubic lattices in dimension D = 3, 4, 5. Taking the continuum limit of the thresholds for discrete hyperspheres we…

Critical p=1/2 in percolation on semi-infinite strips.

  • Z. Koza
  • Mathematics
    Physical review. E
  • 2019
It is argued that the probability that a cluster touches the three sides of a planar system at the percolation threshold has a continuous limit of 1/2 and that this limit is universal for planar systems.

On the monotonicity of the critical time in the Constrained-degree percolation model

Dynamics around the site percolation threshold on high-dimensional hypercubic lattices.

The computational validation employs accelerated random walk simulations with a transfer-matrix description of diffusion to evaluate directly the dynamical critical exponents below d_{u} as well as their logarithmic scaling above d{u}.

N-cluster correlations in four- and five-dimensional percolation

We study N-cluster correlation functions in four- and five-dimensional (4D and 5D) bond percolation by extensive Monte Carlo simulation. We reformulate the transfer Monte Carlo algorithm for…

Bond percolation on simple cubic lattices with extended neighborhoods.

The results show that the percolation thresholds of these and other three-dimensional lattices decrease monotonically with the coordination number z quite accurately according to a power-law p_{c}∼z^{-a} with exponent a=1.111.

Jamming and percolation of k2-mers on simple cubic lattices

A complete analysis of critical exponents and universality has been done, showing that the percolation phase transition involved in the system has the same universality class as the 3D random percolations, regardless of the size k considered.

Random site percolation on honeycomb lattices with complex neighborhoods.

We present a rough estimation-up to four significant digits, based on the scaling hypothesis and the probability of belonging to the largest cluster vs the occupation probability-of the critical…

Percolation of sites not removed by a random walker in d dimensions.

This work systematically explore dependence of the probability Π_{d}(L,u) of percolation (existence of a spanning cluster) of sites not removed by the RW on L and u, which shows the concentration of unvisited sites decays exponentially with increasing u, while the visited sites are highly correlated.

Percolation in a triangle on a square lattice

Percolation on a plane is usually associated with clusters spanning two opposite sides of a rectangular system. Here we investigate three-leg clusters generated on a square lattice and spanning the…

References

SHOWING 1-10 OF 28 REFERENCES

Continuum Percolation Thresholds in Two Dimensions

  • S. MertensC. Moore
  • Computer Science
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2012
This work finds precise values of the percolation transition for disks, squares, rotated squares, and rotated sticks in two dimensions and confirms that these transitions behave as conformal field theory predicts.

Critical percolation in high dimensions.

  • P. Grassberger
  • Computer Science
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2003
Monte Carlo estimates for site and bond percolation thresholds in simple hypercubic lattices with 4-13 dimensions are presented and a scaling law for finite cluster size corrections is proposed.

Corrections to finite size scaling in percolation

A 1 = L-expansion for percolation problems is proposed, where L is the lattice finite length. The square lattice with 27 different sizes L = 18; 22;... 1594 is considered. Certain spanning…

Simultaneous analysis of three-dimensional percolation models

We simulate the bond and site percolation models on several three-dimensional lattices, including the diamond, body-centered cubic, and face-centered cubic lattices. As on the simple-cubic lattice…

LETTER TO THE EDITOR: Precise determination of the critical threshold and exponents in a three-dimensional continuum percolation model

We present a large-scale computer simulation of the prototypical three-dimensional continuum percolation model consisting of a distribution of overlapping (spatially uncorrelated) spheres. By using…

Percolation of overlapping squares or cubes on a lattice

This work proposes a generalization of the excluded volume approximation to discrete systems and uses it to investigate the transition between continuous and discrete percolation, finding a remarkable agreement between the theory and numerical results.

Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses.

The present investigation provides additional analytical results for certain cluster statistics, such as the concentration of k-mers and related quantities, and obtains an upper bound on the percolation threshold η(c), and provides accurate analytical estimates of the pair connectedness function and blocking function for any d as a function of density.

Logarithmic corrections to scaling in critical percolation and random resistor networks.

  • O. StenullH. Janssen
  • Physics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2003
This study comprehends the percolation correlation function, i.e., the probability that two given points are connected, and some of the fractal masses describingpercolation clusters, and calculates the mass of the backbone, the red bonds, and the shortest path.

Recent developments in continuum percolation

Abstract While most natural systems to which percolation theory has been applied are continuum systems, most of the developments in percolation theory have been made through the study of lattice…

Fast Monte Carlo algorithm for site or bond percolation.

  • M. NewmanR. Ziff
  • Physics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2001
An efficient algorithm is described that can measure an observable quantity in a percolation system for all values of the site or bond occupation probability from zero to one in an amount of time that scales linearly with the size of the system.