## Department of Physics Papers

#### Title

#### 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 p_{c} we find that the distribution function depends on percolation exponents for u=n(p_{c}−p)^{Δp} small, where Δ_{p} 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>p_{c} arise from metastable states of the Potts model.

#### Recommended Citation

Harris, A.,
&
Lubensky, T. C.
(1981).
Generalized Percolation.
*Physical Review B,*
*24*
(5),
2656-2670.
http://dx.doi.org/10.1103/PhysRevB.24.2656

**Date Posted:** 12 August 2015

This document has been peer reviewed.