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Number of spanning clusters at the high-dimensional percolation thresholds
Santo Fortunato1, Amnon Aharony, Antonio Coniglio
1Fakultät für Physik, Universität Bielefeld, D-33501 Bielefeld, Germany.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 17, 2004
Summary
A scaling theory predicts how spanning clusters change with lattice size. Simulations confirm predictions in six dimensions, but five dimensions show unexpected scaling behavior.
Area of Science:
- Statistical Mechanics
- Percolation Theory
- Computational Physics
Background:
- Understanding the behavior of systems near critical points is crucial in statistical mechanics.
- Percolation theory studies the formation of connected clusters in random networks.
- The dependence of cluster properties on system size and dimensionality is a fundamental question.
Purpose of the Study:
- To theoretically derive and numerically verify the dependence of the average number of spanning clusters at threshold on lattice size.
- To investigate scaling behaviors in different dimensions, particularly focusing on dimensions d<6, d=6, and d>6.
- To analyze the histogram of spanning cluster multiplicity and its scaling properties.
Main Methods:
- Application of a scaling theory to derive theoretical predictions.
- Numerical simulations to compute the average number of spanning clusters and their multiplicity histograms.
- Analysis of scaling functions and fitting of simulation data to theoretical models.
Main Results:
- Theoretical predictions indicate the average number of spanning clusters becomes independent of lattice size for d<6 and scales as ln L at d=6.
- Simulations in six dimensions align with predictions, including logarithmic corrections.
- In five dimensions, simulations show an unexpected ln L dependence up to L=201, but scaling of the multiplicity histogram suggests a finite average for large L.
- Numerical simulations for d>6 and d=4 are also presented.
Conclusions:
- The study provides insights into the dimensionality dependence of spanning clusters in percolation systems.
- Discrepancies in five dimensions highlight the complexity of scaling behavior and the need for careful analysis of finite-size effects.
- The scaling of the multiplicity histogram offers a robust method to infer the asymptotic behavior of the average number of spanning clusters.
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