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Percolation in finite matching lattices
Stephan Mertens1, Robert M Ziff2
1Institut für Theoretische Physik, Otto-von-Guericke Universität, PF 4120, 39016 Magdeburg, Germany and Santa Fe Institute, 1399 Hyde Park Rd., Santa Fe, New Mexico 87501, USA.
Physical Review. E
|January 14, 2017
Summary
We found a simple formula connecting cluster counts and wrapping probabilities in 2D percolation. This helps accurately estimate the critical density, offering a new view on existing methods.
Area of Science:
- Statistical Mechanics
- Condensed Matter Physics
- Network Science
Background:
- Percolation theory studies the formation of connected clusters in random systems.
- Understanding critical phenomena, like the percolation threshold, is crucial in various scientific fields.
- Existing methods for approximating the percolation threshold have limitations.
Purpose of the Study:
- To derive a new, exact relation for two-dimensional percolation.
- To generalize a classical result by Sykes and Essam.
- To provide a method for accurately determining the critical density.
Main Methods:
- Derivation of an exact mathematical relation.
- Analysis of percolation on periodic lattices of arbitrary size.
- Generalization of existing theoretical frameworks.
Main Results:
- An exact, simple relation between average cluster number and wrapping probabilities.
- The relation is valid for periodic lattices of any size.
- The derived relation facilitates precise approximation of the critical density.
Conclusions:
- The new relation offers a simplified approach to percolation analysis.
- It provides a novel perspective on methods for threshold approximation.
- This work advances the understanding of critical phenomena in 2D systems.
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