Generalized percolation
- 1 September 1981
- journal article
- research article
- Published by American Physical Society (APS) in Physical Review B
- Vol. 24 (5), 2656-2670
- https://doi.org/10.1103/physrevb.24.2656
Abstract
A generalized model of percolation encompassing both the usual model, in which bonds are occupied with probability and are vacant with probability (), and the model appropriate to the statistics of lattice animals, in which the fugacity for occupied bonds is and that for unoccupied bonds is unity, is formulated. Within this model we discuss the crossover between the two problems and we study the statistics of large clusters. We determine the scaling form which the distribution function for the number of clusters with a given number of sites assumes as a function of both and . For near we find that the distribution function depends on percolation exponents for small, where is a crossover exponent, and on exponents appropriate to the lattice-animals problem for large values of . We thus have displayed the relation between the two limits and show conclusively that the lattice-animals exponents cannot be obtained by any simple scaling arguments from the percolation exponents. We also demonstrate that essential singularities in the cluster distribution functions for arise from metastable states of the Potts model.
Keywords
This publication has 22 references indexed in Scilit:
- The critical dimension for lattice animalsJournal of Physics A: General Physics, 1980
- Statistics of lattice animals and dilute branched polymersPhysical Review A, 1979
- Scaling theory of percolation clustersPhysics Reports, 1979
- Field Theory for the Statistics of Branched Polymers, Gelation, and Vulcanization.Physical Review Letters, 1979
- Field Theory for the Statistics of Branched Polymers, Gelation, and VulcanizationPhysical Review Letters, 1978
- Bond percolation processes in d dimensionsJournal of Physics A: General Physics, 1978
- Scaling assumption for lattice animals in percolation theoryJournal of Statistical Physics, 1978
- Percolation processes in d-dimensionsJournal of Physics A: General Physics, 1976
- Lattice animals and percolationJournal of Physics A: General Physics, 1976
- Some Cluster Size and Percolation ProblemsJournal of Mathematical Physics, 1961