We study the site percolation under Achlioptas process (AP) with a product rule in a 2 − dimensional (2D) square lattice. From the measurement of the cluster size distribution, Ps, we find that Ps has a very robust power-law regime followed by a stable hump near the transition threshold. Based on the careful analysis on the Ps distribution, we show that the transition should be discontinuous. The existence of the hysteresis loop in order parameter also verifies that the transition is discontinuous in 2D. Moreover we also show that the transition nature from the product rule is not the same as that from a sum rule in 2D. PACS numbers: 64.60.ah, 64.60.De, 05.70.Fh, 64.60.Bd The percolation transition describing the emergence of large-scale connectivity in lattice systems or complex networks has been extensively studied in statistical mechanics and related fields due to its possible applications to various phenomena such as sol-gel transition and polymerization, resistor networks, and epidemic spreading [1]. When the occupation probability of node (site) is lower than certain threshold p c , all the clusters are microscopic. As the occupation probability increases, the macroscopically connected cluster emerges. Such transition in the ordinary percolation is a continuous transition [1].On the other hand, there have been several attempts to find a percolation model which undergoes a discontinuous transition. The discontinuous percolation transition can be found in the modelling of magnetic systems with significant competition between exchange and crystal-field interactions [2,3]. The similar phenomena has been found in financial systems [4], in which two equally probable phases exit. Other examples of the discontinuous transition in percolation are the formation of infinite cluster under a central-force [5] and the cascade of failure in interdependent networks [6].Recently, Achlioptas et al.[7] suggested a simple process in which the growth of large clusters is systematically suppressed and the process is usually called as Achlioptas process (AP). Based on the analysis of transition interval it was argued that the percolation transition under AP is explosive and discontinuous. Several variant of models have been investigated to understand the general properties and conditions which cause such non-trivial discontinuous transition [8][9][10][11]. Some examples of such non-trivial transition has been found in nano-tube based system [11], protein homology network [12], and community formation [13].However, more recent studies on the percolation transition under AP reveals several evidences which strongly suggest that the transition can be continuous. For example, da Costa et al. [14] argued that the transition in the complete graph (CP) is continuous, even though the order parameter exponent is very small (β ≃ 0.056). From the measurement of the cluster size distribution Lee et al.[15] also argued that the transition in CP is continuous. Grassberger et al.[16] also argued that the transition, even in the low-dimension...

We investigate the kinetics of bimolecular chemical reactions A+A→0 and A+B→0 on weighted scale-free networks (WSFNs) with degree distribution P(k)∼k^{-γ} . On WSFNs, a weight w{ij} is assigned to the link between node i and j . We consider the symmetric weight given as w{ij}=(k{i}k{j})^{μ} , where k{i} and k{j} are the degree of node i and j . The hopping probability T{ij} of a particle from node i to j is then given as T{ij}∝(k{i}k{j})^{μ} . From a mean-field analysis, we analytically show in the thermodynamic limit that the kinetics of A+A→0 and A+B→0 are identical and there exist two crossover μ values, μ{1c}=γ-2 and μ{2c}=(γ-3)/2 . The density of particles ρ(t) algebraically decays in time t as t^{-α} with α=1 for μ<μ{2c} and α=(μ+1)/(γ-μ-2) for μ{2c}≤μ<μ{1c} . For μ≥μ{1c} , ρ decays exponentially. With the mean-field rate equation for ρ(t) , we also analytically show that the kinetics on the WSFNs is mapped onto that on unweighted SFNs with P(k)∼k^{-γ^{'}} with γ^{'}=(μ+γ)/(μ+1) .

To establish the bond-site duality of explosive percolations in two dimensions, the site and bond explosive-percolation models are carefully defined on a square lattice. By studying the cluster distribution function and the behavior of the second largest cluster, it is shown that the duality in which the transition is discontinuous exists for the pairs of the site model and the corresponding bond model which relatively enhances the intrabond occupation. In contrast the intrabond-suppressed models which have no corresponding site models undergo a continuous transition and satisfy the normal scaling ansatz as ordinary percolation.