## Department of Physics Papers

#### Document Type

Journal Article

#### Date of this Version

9-1-1983

#### Publication Source

Physical Review B

#### Volume

28

#### Issue

5

#### Start Page

2614

#### Last Page

2629

#### DOI

10.1103/PhysRevB.28.2614

#### Abstract

A general method is given whereby m-connectedness correlation functions can be studied in the percolation problem. The method for m=1 involves calculating the *k*th power of the correlation function ⟨σ(x)σ(x′)⟩ for a randomly dilute Ising model at *nonzero temperature* and subsequently averaging over configurations. The final step is to take the limit k→0. This method is tested by reproducing the standard results for the percolation problem from an extension of the calculation of Stephen and Grest. For m>1 an additional "color" index is introduced and a Hamiltonian is constructed in which different colors repel one another, thereby giving an exact prescription for m-connectedness. Order parameters for m-connectedness are identified. The m=2 order parameter couples through a trilinear term to the m=1 order parameter. The main result is that β(m) the exponent for m-connectedness is given by β^{(m)}=mβ+νψ^{(m)}, where β and ν are the usual exponents for percolation and ψ^{(m)} is a new crossover exponent which, to lowest order in ε=6−d, is given by ψ^{(m)}=m(m−1)ε^{2}/49. This result implies that the fractal dimensionality of the *biconnected* part of the critically percolating cluster is given in terms of the percolation critical exponents as γ/ν−ψ^{(2)}. If "nodes" are defined as triconnected points, then β^{(3)} is the critical exponent associated with their density in the infinite cluster. We also discuss evidence that the "node-link" model of Skal-Shklovskii-de Gennes breaks down for d less than some critical value d^{^}. Numerically we adduce evidence that d^{^} may be larger than 3.

#### Recommended Citation

Harris, A.
(1983).
Field-Theoretic Approach to Biconnectedness in Percolating Systems.
*Physical Review B,*
*28*
(5),
2614-2629.
http://dx.doi.org/10.1103/PhysRevB.28.2614

**Date Posted:** 12 August 2015

This document has been peer reviewed.

## Comments

At the time of publication, author A. Brooks Harris was affiliated with Sclumberger-Doll Research, Ridgefield, Connecticut. Currently, (s)he is a faculty member in the Department of Physics at the University of Pennsylvania.