Site-bond percolation: a low-density series study of the uncorrelated limit

P Agrawal, S Redner, P J Reynolds, H E Stanley
1979 Journal of Physics A: Mathematical and General  
A generalisation of the pure site and pure bond percolation problems is studied, in which both the sites and bonds are independently occupied at random. This generalisation-the site-bond problem-is of current interest because of its application to the phenomenon of polymer gelation. Motivated by considerations of cluster connectivity, we have defined two distinct models for site-bond percolation, models A and B. In model A, a cluster is considered to be a set of occupied bonds and sites in
more » ... the bonds are joined by occupied sites, and the sites are joined by occupied bonds. Since a bond cannot contribute to cluster connectivity if either site at its endpoints is not occupied, we define model B in which these 'non-connecting' bonds are treated as part of the cluster perimeter. We prove that the critical curve and critical exponents are the same for both models. For model B, we calculate low-density series expansions for the mean cluster size on the square lattice. We calculate three different series, using the following definitions of cluster size: site size, bond size, and a hybrid measure involving both site and bond size. All three series have been used to obtain the phase boundary between the percolating (gel) and non-percolating (sol) regions. Numerical evidence is presented which indicates that along the entire phase boundary the mean-size exponent y assumes a universal value.
doi:10.1088/0305-4470/12/11/018 fatcat:mjsxdmptfrf7jj6pg3qylxya3u