Critical percolation clusters in seven dimensions and on a complete graph

We study critical bond percolation on a seven-dimensional hypercubic lattice with 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 $n(s,V)\ensuremath{\sim}{s}^{\ensuremath{-}\ensuremath{\tau}}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{n}(s/{V}^{{d}_{\mathrm{f}}^{*}})$ with identical volume fractal dimension ${d}_{\mathrm{f}}^{*}=2/3$ and exponent $\ensuremath{\tau}=1+1/{d}_{\mathrm{f}}^{*}=5/2$. 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, ${\ensuremath{\rho}}_{\mathrm{n},\mathrm{CG}}\ensuremath{\rightarrow}0$, for large CGs, but converges to a finite value, ${\ensuremath{\rho}}_{\mathrm{n},7\mathrm{D}}=0.006\phantom{\rule{0.16em}{0ex}}193\phantom{\rule{0.16em}{0ex}}1(7)$, for the 7D hypercube. Further, we observe that while the bridge-free dimension ${d}_{\mathrm{bf}}^{*}=1/3$ holds for both the CG and 7D cases, the volume fractal dimensions of the leaf-free clusters are different: ${d}_{\mathrm{lf},7\mathrm{D}}^{*}=0.669(9)\ensuremath{\approx}2/3$ and ${d}_{\mathrm{lf},\mathrm{CG}}^{*}=0.3337(17)\ensuremath{\approx}1/3$. On the CG and in 7D, the whole, leaf-free, and bridge-free clusters all have the shortest-path volume fractal dimension ${d}_{\mathrm{min}}^{*}\ensuremath{\approx}1/3$, 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 $\ensuremath{\sim}lnV$, while for 7D they scale as $\ensuremath{\sim}V$. For the size distribution, we find the behavior on the CG is governed by a modified Fisher exponent ${\ensuremath{\tau}}^{\ensuremath{'}}=1$, while for leaf-free clusters in 7D, it is governed by Fisher exponent $\ensuremath{\tau}=5/2$. The size distribution of bridge-free clusters in 7D displays two-scaling behavior with exponents $\ensuremath{\tau}=4$ and ${\ensuremath{\tau}}^{\ensuremath{'}}=1$. The probability distribution $P({C}_{1},V)d{C}_{1}$ of the largest cluster of size ${C}_{1}$ for whole percolation configurations is observed to follow a single-variable function $\overline{P}(x)dx$, with $x\ensuremath{\equiv}{C}_{1}/{V}^{{d}_{\mathrm{f}}^{*}}$ 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 $x\ensuremath{\rightarrow}0$ and $x\ensuremath{\rightarrow}\ensuremath{\infty}$ 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.