Critical percolation clusters in seven dimensions and on a complete graph

Abstract

We study critical bond percolation on a seven-dimensional (7D) hypercubic lattice with periodic boundary conditions and on the complete graph (CG) of finite volume $V$. We numerically confirm that for both cases, the critical number density $n(s,V)$ of clusters of size $s$ obeys a scaling form $n(s,V) \sim s^{-τ} \tilde{n} (s/V^{d^*_{\rm f}})$ with identical volume fractal dimension $d^*_{\rm f}=2/3$ and exponent $τ= 1+1/d^*_{\rm f}=5/2$. We then classify occupied bonds into {\em bridge} bonds, which includes {\em branch} and {\em junction} bonds, and {\em non-bridge} 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 {\em leaf-free} configurations, whereas, deleting all bridge bonds leads to {\em bridge-free} configurations. It is shown that the fraction of non-bridge (bi-connected) bonds vanishes $ρ_{\rm n, CG}$$\rightarrow$0 for large CGs, but converges to a finite value $ ρ_{\rm n, 7D} =0.006 \, 193 \, 1(7)$ for the 7D hypercube. Further, we observe that while the bridge-free dimension $d^*_{\rm bf}=1/3$ holds for both the CG and 7D cases, the volume fractal dimensions of the leaf-free clusters are different: $d^*_{\rm \ell f, 7D} = 0.669 (9) \approx 2/3$ and $d^*_{\rm \ell f, CG} = 0. 333 7 (17) \approx 1/3$. 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 $\sim \ln V$, while for 7D they scale as $\sim V$. Our work demonstrates that the geometric structure of high-dimensional percolation clusters cannot be fully accounted for by their complete-graph counterparts.