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 Z(R) and Z(Q) for SR-EI (defined by R-(x()) and Q corresponding to two different p-values, respectively), we determine the thresholds p(c) of all models with best precision. We also estimate the critical exponents and beta/nu and gamma/nu for PR-SI, SR-SI and PR-EI by studying the critical behaviors of the size of the largest cluster C-1 and the second moment M-2 of sizes of all clusters. For SR-EI, from C-1 and M-2 we only obtain pseudo-critical exponents, which are nonphysical. Precisely at p(c), 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) similar to s(-tau)(') (n) over tilde (s/L-d'), where tau' = 1 + d/d' with d' = d(F) (fractal dimension) for PR-SI, SR-SI, PR-EI, d' = d (spatial dimension) for SR-EI, and (n) over tilde (x) with x = s/L-d' is an universal scaling function. Based on critical cluster-size distribution, we conjecture the values of beta/nu (and gamma/nu 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.