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Updated: Jul 6, 2026

Planar Gradient Diffusion System to Investigate Chemotaxis in a 3D Collagen Matrix

Planar Gradient Diffusion System to Investigate Chemotaxis in a 3D Collagen Matrix

Published on: June 12, 2015

Site percolation on planar Phi(3) random graphs.

¹Laboratoire de Physique Théorique et Modélisation, CNRS-Université de Cergy-Pontoise-UMR8089, Cergy-Pontoise Cedex, France. kownacki@ptm.u-cergy.fr

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|March 21, 2008

Summary

Site percolation on random Phi(3) planar graphs was studied. Monte Carlo simulations revealed a percolation threshold of p(c) = 0.7360(5), with critical exponents aligning with bond percolation.

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Last Updated: Jul 6, 2026

Planar Gradient Diffusion System to Investigate Chemotaxis in a 3D Collagen Matrix

Planar Gradient Diffusion System to Investigate Chemotaxis in a 3D Collagen Matrix

Published on: June 12, 2015

Area of Science:

Graph theory
Statistical physics
Computational science

Background:

Percolation theory investigates the formation of connected clusters in random systems.
Random Phi(3) planar graphs provide a complex network structure for studying percolation phenomena.
Understanding percolation thresholds is crucial for various applications, from material science to network analysis.

Purpose of the Study:

To investigate site percolation on random Phi(3) planar graphs.
To determine the percolation threshold (p(c)) for this specific graph type.
To compare the critical exponents with known values from bond percolation.

Main Methods:

Utilized Monte Carlo numerical techniques for simulations.
Generated random Phi(3) planar graphs.
Randomly removed vertices (fraction q = 1-p) to form clusters of occupied sites.
Measured properties of cluster distribution to identify the percolation threshold.

Main Results:

Percolation was observed to occur at an occupation probability above a critical threshold.
The determined percolation threshold was found to be p(c) = 0.7360(5).
The calculated critical exponents were found to be compatible with those analytically known for bond percolation.

Conclusions:

Site percolation on random Phi(3) planar graphs exhibits a distinct percolation threshold.
The findings suggest a universality class for this type of percolation that aligns with bond percolation.
Monte Carlo simulations are effective for studying complex network phenomena like percolation.