Department of Physics Papers

Document Type

Journal Article

Date of this Version

9-1-1981

Publication Source

Physical Review B

Volume

24

Issue

5

Start Page

2656

Last Page

2670

DOI

10.1103/PhysRevB.24.2656

Abstract

A generalized model of percolation encompassing both the usual model, in which bonds are occupied with probability p and are vacant with probability (1−p), and the model appropriate to the statistics of lattice animals, in which the fugacity for occupied bonds is p 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 n assumes as a function of both n and p. For p near pc we find that the distribution function depends on percolation exponents for u=n(pc−p)Δp small, where Δp is a crossover exponent, and on exponents appropriate to the lattice-animals problem for large values of u. 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 p>pc arise from metastable states of the Potts model.

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Date Posted: 12 August 2015

This document has been peer reviewed.