Series expansion analysis of the backbone properties of two-dimensional percolation clusters

Abstract

Low-density series expansions for the backbone properties of two-dimensional bond percolation clusters are derived and analysed. Expansions for most of the 14 properties considered are new and are obtained to order on the square lattice and order on the triangular lattice. Earlier series work was confined to three properties of the square lattice and was to order . The fractal dimension of the bonds or sites in the backbone is estimated to be and is intermediate between a previously conjectured field theory value and the latest Monte Carlo results. The union, intersection and length of the longest self-avoiding paths are found to have the same fractal dimension which is close to and consistent with the field theory conjecture for . On the other hand, the union intersection and length of the shortest paths are found to have different dimensions and in the case of the intersection, the triangular and square lattices are found to have sigificantly different dimensions. The fractal dimension of the shortest path also appears to be non-universal and we find for the square lattice and for the triangular lattice. Critical amplitude ratios are considered and found to be in agreement with theoretical inequalities.