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… 

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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.