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Physical review. B, Condensed matter

Field-theoretic approach to biconnectedness in percolating systems

A. Brooks Harris

Published Sep 1, 1983• 79 citations
Publisher

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 kth power of the correlation function $ 〈\ensuremath{\sigma}(x)\ensuremath{\sigma}({x}^{\ensuremath{'}})〉 $ for a randomly dilute Ising model at nonzero temperature and subsequently averaging over configurations. The final step is to take the limit $k\ensuremath{\rightarrow}0$. 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 ${\ensuremath{\beta}}^{(m)}$ the exponent for $m$-connectedness is given by ${\ensuremath{\beta}}^{(m)}=m\ensuremath{\beta}+\ensuremath{\nu}{\ensuremath{\psi}}^{(m)}$, where $\ensuremath{\beta}$ and $\ensuremath{\nu}$ are the usual exponents for percolation and ${\ensuremath{\psi}}^{(m)}$ is a new crossover exponent which, to lowest order in $\ensuremath{\epsilon}=6\ensuremath{-}d$, is given by ${\ensuremath{\psi}}^{(m)}=\frac{m(m\ensuremath{-}1){\ensuremath{\epsilon}}^{2}}{49}$. 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 $\frac{\ensuremath{\gamma}}{\ensuremath{\nu}}\ensuremath{-}{\ensuremath{\psi}}^{(2)}$. If "nodes" are defined as triconnected points, then ${\ensuremath{\beta}}^{(3)}$ 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 $\stackrel{^}{d}$. Numerically we adduce evidence that $\stackrel{^}{d}$ may be larger than 3.

DOI
10.1103/physrevb.28.2614

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