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

Abstract

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 [Formula: see text] and a ratio [Formula: see text] for PR-SI, SR-SI, PR-EI, and the composite quantities [Formula: see text] and [Formula: see text] for SR-EI (defined by [Formula: see text] and [Formula: see text] corresponding to two different [Formula: see text]-values, respectively), we determine the thresholds [Formula: see text] of all models with best precision. We also estimate the critical exponents [Formula: see text] and [Formula: see text] for PR-SI, SR-SI and PR-EI by studying the critical behaviors of the size of the largest cluster [Formula: see text] and the second moment [Formula: see text] of sizes of all clusters. For SR-EI, from [Formula: see text] and [Formula: see text], we only obtain pseudo-critical exponents, which are nonphysical. Precisely at [Formula: see text], we study the critical cluster-size distribution [Formula: see text] (number density of the clusters of size [Formula: see text]) for all models and find that it can be described by [Formula: see text], where [Formula: see text] with [Formula: see text] (fractal dimension) for PR-SI, SR-SI, PR-EI, [Formula: see text] (spatial dimension) for SR-EI, and [Formula: see text] with [Formula: see text] is an universal scaling function. Based on critical cluster-size distribution, we conjecture the values of [Formula: see text] (and [Formula: see text] 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.