Critical percolation clusters in seven dimensions and on a complete graph

We study critical bond percolation on a seven-dimensional hypercubic latticewith periodic boundary conditions (7D) and on the complete graph (CG) of finite volume (number of vertices) V. We numerically confirm that for both cases, the critical number density n(s, V) of clusters of size s obeys a scaling form n(s, V) similar to s(-tau) (n) over tilde (s/V-d*(f)) with identical volume fractal dimension d*(f) = 2/3 and exponent tau = 1 + 1/d*(f) = 5/2. We then classify occupied bonds into bridge bonds, which includes branch and junction bonds, and nonbridge bonds; a bridge bond is a branch bond if and only if its deletion produces at least one tree. Deleting branch bonds from percolation configurations produces leaf-free configurations, whereas deleting all bridge bonds leads to bridge-free configurations composed of blobs. It is shown that the fraction of nonbridge (biconnected) bonds vanishes, rho(n),(CG) -> 0, for large CGs, but converges to a finite value, rho(n, 7D) = 0.006 193 1(7), for the 7D hypercube. Further, we observe that while the bridge-free dimension d*(bf) = 1/3 holds for both the CG and 7D cases, the volume fractal dimensions of the leaf-free clusters are different: d*(lf, 7D) = 0.669(9) approximate to 2/3 and d*(lf, CG) = 0.3337(17) approximate to 1/3. On the CG and in 7D, the whole, leaf-free, and bridge-free clusters all have the shortest-path volume fractal dimension d*(min) approximate to 1/3, characterizing their graph diameters. We also study the behavior of the number and the size distribution of leaf-free and bridge-free clusters. For the number of clusters, we numerically find the number of leaf-free and bridge-free clusters on the CG scale as similar to ln V, while for 7D they scale as similar to V. For the size distribution, we find the behavior on the CG is governed by a modified Fisher exponent tau' = 1, while for leaf-free clusters in 7D, it is governed by Fisher exponent tau = 5/2. The size distribution of bridge-free clusters in 7D displays two-scaling behavior with exponents tau = 4 and tau' = 1. The probability distribution P(C-1, V)dC(1) of the largest cluster of size C1 forwhole percolation configurations is observed to followa single-variable function (P) over bar (x)dx, with x = C-1/V-d*(f) for both CG and 7D. Up to a rescaling factor for the variable x, the probability functions for CG and 7D collapse on top of each other within the entire range of x. The analytical expressions in the x -> 0 and x -> infinity limits are further confirmed. Our work demonstrates that the geometric structure of high-dimensional percolation clusters cannot be fully accounted for by their complete-graph counterparts.