Thresholds and critical exponents of explosive bond percolation on the square lattice

In this paper, we present a Monte Carlo study of four explosive bond percolation models on the square lattice: (i) product rule which suppresses intrabonds (PR-SI), (ii) sum rule which suppresses intrabonds (SR-SI), (iii) product rule which enhances intrabonds (PR-EI) and (iv) sum rule which enhances intrabonds (SR-EI). By performing extensive simulations and finite-size scaling analysis of the wrapping probability R(x) and a ratio Q for PR-SI, SR-SI, PR-EI, and the composite quantities ZR and ZQ for SR-EI (defined by R(x) and Q corresponding to two different p-values, respectively), we determine the thresholds pc of all models with best precision. We also estimate the critical exponents β∕ν and γ∕ν for PR-SI, SR-SI and PR-EI by studying the critical behaviors of the size of the largest cluster C1 and the second moment M2 of sizes of all clusters. For SR-EI, from C1 and M2, we only obtain pseudo-critical exponents, which are nonphysical. Precisely at pc, we study the critical cluster-size distribution n(s,L) (number density of the clusters of size s) for all models and find that it can be described by n(s,L)∼s−τ′ñ(s∕Ld′), where τ′=1+d∕d′ with d′=dF (fractal dimension) for PR-SI, SR-SI, PR-EI, d′=d (spatial dimension) for SR-EI, and ñ(x) with x=s∕Ld′ is an universal scaling function. Based on critical cluster-size distribution, we conjecture the values of β∕ν (and γ∕ν with the help of a scaling relation) for SR-EI. It is found that the exponents for PR-SI and SR-SI are consistent with each other, but the ones for PR-EI and SR-EI are different. Our results disclose two facts: (1) all models investigated here undergo continuous phase transitions, since their behaviors can be described by typical scaling formulas for continuous phase transitions; (2) PR-SI and SR-SI belong to a same universality class, however, PR-EI and SR-EI belong to different universality classes, and all of them differ from which random percolation belongs to. This work provides a testing ground for future theoretical studies.