Field theoretic approaches to biconnectedness in percolating systems

Abstract

Two field theoretic formulations for the percolation problem are presented from which the critical exponents describing the 'backbone' of the infinite cluster at the percolation threshold are obtained. At high spatial dimension, d, the order-parameter exponent for the backbone beta ⁽²⁾ is given by beta ⁽²⁾=2 beta + psi ⁽²⁾ nu , where beta is the critical exponent for the density of the infinite cluster and psi ⁽²⁾ is a new crossover exponent. In mean-field theory psi ⁽²⁾=0 and for d=6- epsilon , psi ⁽²⁾=2 epsilon ²/49+O( epsilon ³). Presumably, psi ⁽²⁾(d) is a smooth function of d for d>d*, where numerical and theoretical work indicates that d* is about 3. The result indicates that the fractal dimensionality of the backbone is given in terms of the percolation exponents as gamma / nu - psi ⁽²⁾.