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Large clusters in supercritical percolation.

¹A. V. Luikov Heat and Mass Transfer Institute, National Academy of Sciences of Belarus, 15 P. Brovka Street, Minsk 220072, Belarus.

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

|September 21, 2002

Summary

Large clusters in supercritical percolation become compact above the threshold. Cluster surface area significantly influences their statistical behavior, impacting decay laws.

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Area of Science:

Statistical physics
Percolation theory

Background:

Investigating large finite clusters in supercritical percolation on finite lattices below the critical dimension (d(c)=6).
Understanding cluster size statistics and decay laws is crucial in various physical systems.

Purpose of the Study:

To investigate the statistical behavior of large finite clusters in supercritical percolation.
To establish the correlation between critical exponents (zeta) and cluster surface properties.
To find corrections to the cluster decay law for compact clusters.

Main Methods:

Derivation of an approximate system of ordinary differential equations for finite clusters.
Analysis of the relationship between cluster size, surface area, and decay exponents.
Numerical testing using Monte Carlo simulations on 2D and 3D lattices.

Main Results:

A correlation is found between critical exponents (zeta) and cluster surface area.
For clusters without self-intersections, zeta=1; for those with few, zeta=1-eta.
The first correction to the cluster decay law for compact clusters was determined.

Conclusions:

Above the percolation threshold, most large clusters are compact.
Cluster surface area is a primary factor governing behavior in supercritical percolation.
The findings are supported by numerical simulations on 2D and 3D lattices.

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