Beyond blobs in percolation cluster structure: the distribution of 3-blocks at the percolation threshold

The incipient infinite cluster appearing at the bond percolation threshold can be decomposed into singly connected "links" and multiply connected "blobs." Here we decompose blobs into objects known in graph theory as 3-blocks. A 3-block is a graph that cannot be separated into disconnected subgraphs by cutting the graph at two or fewer vertices. Clusters, blobs, and 3-blocks are special cases of k-blocks with k=1, 2, and 3, respectively. We study bond percolation clusters at the percolation threshold on two-dimensional (2D) square lattices and three-dimensional cubic lattices and, using Monte Carlo simulations, determine the distribution of the sizes of the 3-blocks into which the blobs are decomposed. We find that the 3-blocks have fractal dimension d(3)=1.2+/-0.1 in 2D and 1.15+/-0.1 in 3D. These fractal dimensions are significantly smaller than the fractal dimensions of the blobs, making possible more efficient calculation of percolation properties. Additionally, the closeness of the estimated values for d(3) in 2D and 3D is consistent with the possibility that d(3) is dimension independent. Generalizing the concept of the backbone, we introduce the concept of a "k-bone," which is the set of all points in a percolation system connected to k disjoint terminal points (or sets of disjoint terminal points) by k disjoint paths. We argue that the fractal dimension of a k-bone is equal to the fractal dimension of k-blocks, allowing us to discuss the relation between the fractal dimension of k-blocks and recent work on path crossing probabilities.