Field-Theoretic Approach to Biconnectedness in Percolating Systems

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 ⟨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)ε2/49. 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.