The BFW model introduced by Bohman, Frieze, and Wormald [Random Struct. Algorithms, 25, 432 (2004)] and recently investigated in the framework of discontinuous percolation by Chen and D'Souza [Phys. Rev. Lett., 106, 115701 (2011)], is studied on the square and simple-cubic lattices. In two and three dimensions, we find numerical evidence for a strongly discontinuous transition. In two dimensions, the clusters at the threshold are compact with a fractal surface of fractal dimension $d_f=1.49\pm0.02$. On the simple-cubic lattice, distinct jumps in the size of the largest cluster are observed. We proceed to analyze the tree-like version of the model, where only merging bonds are sampled, for dimension two to seven. The transition is again discontinuous in any considered dimension. Finally, the dependence of the cluster-size distribution at the threshold on the spatial dimension is also investigated.
Percolation is a classical model in Statistical Physics which, despite consisting of simple geometrical rules, has found application in a broad range of problems [1,2]. In classical (random) percolation, a fraction p of sites or bonds are occupied uniformly at random. At a critical occupation fraction p c the system undergoes a continuous transition from a non-percolating to a percolating state. A recent work by Achlioptas, D'Souza, and Spencer [3] hinted at the possibility of a discontinuous percolation transition by slightly modifying the bond selection rules of evolving graphs, coining the term explosive percolation. This surprising result led to intensive research efforts studying the model on different topologies and dimensions, yielding ambiguous results regarding the nature of the transition [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. More recently, the transition in the original model was formally demonstrated to be continuous in the mean-field limit [19] and there is evidence that the same applies to other models [20][21][22][23]. Alternative models have been devised and analyzed in detail [24][25][26][27][28][29][30][31][32][33], several of them with clear signs supporting the hypothesis of a discontinuous transition.
In particular, Chen and D’Souza considered the percolation properties on the random graph of the BFW model [34,35], originally introduced by Bohman, Frieze, and Wormald [36], reporting a discontinuous transition with several giant components in the thermodynamic limit. In general, the behavior of a system undergoing a transition depends on the dimension [37]. In this paper, we analyze the original BFW model on the 2D square lattice and on the 3D cubic lattice, as well as the tree-like version up to dimension seven.
The manuscript is organized as follows. In Sec. II we discuss the BFW model, adapted to the square lattice, along with some definitions. Section III contains the extension to three dimensions. The tree-like version is analyzed in Sec. IV. We conclude with the final remarks in Sec. V.
The BFW model has recently been analyzed by Chen and D’Souza [34,35] on the random graph, by extending the algorithm initially introduced by Bohman, Frieze, and Wormald [36]. Here, we consider this model on a square lattice of linear size L with periodic boundary conditions and number of sites N , where N = L 2 . Initially, all sites are occupied and all bonds are unoccupied (empty), such that there are N clusters of unitary size. As in Ref. [34,35], one occupies the bonds according to the following procedure. Let u be the total number of selected bonds, t the number of occupied bonds, and k the stage of the process, initially set to k = 2. The first bond is occupied at random, such that t = u = 1. Then, at each step u, 1. One bond, among the unoccupied ones, is selected uniformly at random. [34,36] for the random graph, k is essentially equal to smax. Therefore the corresponding curves in the figure, P∞ and k/N , almost overlap. The upper inset shows a close-up view of the upper-left corner of the main plot, as indicated by the box. The system size is N = 512 2 , the measured values are averages over more than 10 5 samples. In the lower inset we see the same variables as the main plot, measured for a single realization.
- If l ≤ k, go to (4). Else if t/u ≥ g(k) = 1/2 + 1/(2k), go to (5).
Else increment k by one and go to (3).
- Occupy the selected bond.
Note that the above procedure, in (1), only samples unoccupied bonds, while in Ref. [34][35][36] edges are chosen uniformly at random from all edges, whether occupied or not. In adapting random graph models to the lattice, we solely sample unoccupied bonds since in classical (random) percolation the control parameter is the fraction of occupied bonds [4,5,7,8,21]. In contrast, for the BFW model on the random graph, by sampling all bonds, since sufficiently many new links can be accepted asymptotically, more than one stable macroscopic cluster is obtained [34,35]. The procedure is applied iteratively while the system evolves from t = 0, corresponding to a bond occupation fraction of p = t/(2N ) = 0, to t = 2N where all bonds in the system are occupied, i. e., p = 1. By construction, at each stage, the size of the largest cluster can be at most s max ≤ k (see Fig. 1) [36]. To illustrate the evolution of this process, we monitor in Fig. 1 the usual order parameter for the percolation transition, defined as the fraction of sites in the largest cluster P ∞ = s max /N . One observes that the transition is delayed, compared to classical percolation where p c = 1/2, and the growth of the order parameter at the threshold appears to be more pronounced than in the classical case (numerical evidence discussed below). In addition, Fig. 1 shows the behavior of the stage per site k/N , which grows as the system evolves and which imposes a bound on the order parameter, P ∞ ≤ k/N . The fraction of accepted bonds t/u and the value of g(k)
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