Scaling at the percolation threshold above six dimensions

Geometric scaling behaviors of the Fortuin-Kasteleyn Ising model in high dimensions

Recently, we argued [Chin. Phys. Lett. 39, 080502 (2022)0256-307X10.1088/0256-307X/39/8/080502] that the Ising model simultaneously exhibits two upper critical dimensions (d_{c}=4,d_{p}=6) in the Fortuin-Kasteleyn (FK) random-cluster representation. In this paper, we perform a systematic study of the FK Ising model on hypercubic lattices with spatial dimensions d from 5 to 7, and on the complete graph. We provide a detailed data analysis of the critical behaviors of a variety of quantities at and near the critical points. Our results clearly show that many quantities exhibit distinct critical phenomena for 4<d<6 and d≥6, and thus strongly support the argument that 6 is also an upper critical dimension. Moreover, for each studied dimension, we observe the existence of two configuration sectors, two lengthscales, as well as two scaling windows, and thus two sets of critical exponents are needed to describe these behaviors. Our finding enriches the understanding of the critical phenomena in the Ising model.

Dynamics around the site percolation threshold on high-dimensional hypercubic lattices

Recent advances on the glass problem motivate reexamining classical models of percolation. Here we consider the displacement of an ant in a labyrinth near the percolation threshold on cubic lattices both below and above the upper critical dimension of simple percolation, d_{u}=6. Using theory and simulations, we consider the scaling regime and obtain that both caging and subdiffusion scale logarithmically for d≥d_{u}. The theoretical derivation, which considers Bethe lattices with generalized connectivity and a random graph model, confirms that logarithmic scalings should persist in the limit d→∞. The computational validation employs accelerated random walk simulations with a transfer-matrix description of diffusion to evaluate directly the dynamical critical exponents below d_{u} as well as their logarithmic scaling above d_{u}. Our numerical results improve various earlier estimates and are fully consistent with our theoretical predictions.

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)∼s^{-τ}n[over ̃](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}→0, for large CGs, but converges to a finite value, ρ_{n,7D}=0.0061931(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)≈2/3 and d_{lf,CG}^{*}=0.3337(17)≈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}^{*}≈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 ∼lnV, while for 7D they scale as ∼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)dC_{1} of the largest cluster of size C_{1} for whole percolation configurations is observed to follow a single-variable function P[over ¯](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→∞ 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.

Network Anatomy Controlling Abrupt-like Percolation Transition

We virtually dissect complex networks in order to understand their internal structure, just as doctors do with the bodies of animals. Our novel method classifies network links into four categories: bone, fat, cartilage, and muscle, based on network connectivity. We derive an efficient percolation strategy from this new viewpoint of network anatomy, which enables abrupt-like percolation transition through removal of a small amount of cartilage links, which play a crucial role in network connectivity. Furthermore, we find nontrivial scaling laws in the relationships between four types of links in each cluster and evaluate power exponents, which characterize network structures as seen in the real large-scale network of trading business firms and in the Erdős-Rényi network. Finally, we observe changes in the transition point for random bond percolation process, demonstrating that the addition of muscle links enhances network robustness, while fat links are irrelevant. These findings aid in controlling the percolation transition for an arbitrary network.

Nonequilibrium phase transitions and finite-size scaling in weighted scale-free networks

We consider nonequilibrium phase transitions, such as epidemic spreading, in weighted scale-free networks, in which highly connected nodes have a relatively smaller ability to transfer infection. We solve the dynamical mean-field equations and discuss finite-size scaling theory. The theoretical predictions are confronted with the results of large scale Monte Carlo simulations on the weighted Barabási-Albert network. Local scaling exponents are found different at a typical site and at a node with very large connectivity.

Number of spanning clusters at the high-dimensional percolation thresholds

A scaling theory is used to derive the dependence of the average number k of spanning clusters at threshold on the lattice size L. This number should become independent of L for dimensions d<6 and vary as ln L at d=6 . The predictions for d>6 depend on the boundary conditions, and the results there may vary between L(d-6) and L0. While simulations in six dimensions are consistent with this prediction [after including corrections of order ln(ln L)], in five dimensions the average number of spanning clusters still increases as ln L even up to L=201 . However, the histogram P(k) of the spanning cluster multiplicity does scale as a function of kX(L), with X(L) =1+const/L, indicating that for sufficiently large L the average k will approach a finite value: a fit of the five-dimensional multiplicity data with a constant plus a simple linear correction to scaling reproduces the data very well. Numerical simulations for d>6 and for d=4 are also presented.