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Mapping functions and critical behavior of percolation on rectangular domains.
Hiroshi Watanabe1, Chin-Kun Hu
1Department of Complex Systems Science, Graduate School of Information Science, Nagoya University, Furouchou, Chikusa-ku, Nagoya 464-8601, Japan. hwatanabe@is.nagoya-u.ac.jp
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 13, 2008
Summary
Researchers studied bond percolation on rectangular domains, finding that superscaling behavior of existence probability and percolation probability can be explained by mapping functions. These functions allow calculation of key exponents.
Area of Science:
- Statistical physics
- Complex systems
- Percolation theory
Background:
- Percolation theory studies the formation of connected clusters in random networks.
- Rectangular domains with varying aspect ratios present unique challenges in percolation analysis.
- Previous work identified superscaling behavior in existence and percolation probabilities.
Purpose of the Study:
- To investigate the existence probability (Ep) and percolation probability (P) in bond percolation on rectangular domains.
- To analyze the influence of domain aspect ratio (R) on these probabilities.
- To explain the observed superscaling behavior using mapping functions.
Main Methods:
- Utilizing mapping functions to relate systems with different aspect ratios.
- Analyzing lower-order approximations of mapping functions (f_R for Ep, g_R for P).
- Determining exponents 'a' and 'b' from numerically computed mapping functions.
Main Results:
- The mapping functions provide an understanding of the superscaling behavior of Ep and P.
- Exponents 'a' and 'b' governing superscaling can be derived from these mapping functions.
- The study confirms and explains previously observed superscaling phenomena.
Conclusions:
- Mapping functions are crucial for understanding percolation phenomena in systems with varying aspect ratios.
- The derived exponents 'a' and 'b' quantify the superscaling behavior.
- This approach offers a unified framework for analyzing percolation on diverse rectangular domains.
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