Nonlocal product rules for percolation.
Despite original claims of a first-order transition in the product rule model proposed by Achlioptas et al. [Science 323, 1453 (2009)], recent studies indicate that this percolation model, in fact, displays a continuous transition. The distinctive scaling properties of the model at criticality, however, strongly suggest that it should belong to a different universality class than ordinary percolation. Here we introduce a generalization of the product rule that reveals the effect of nonlocality on the critical behavior of the percolation process. Precisely, pairs of unoccupied bonds are chosen according to a probability that decays as a power law of their Manhattan distance, and only that bond connecting clusters whose product of their sizes is the smallest becomes occupied. Interestingly, our results for two-dimensional lattices at criticality shows that the power-law exponent of the product rule has a significant influence on the finite-size scaling exponents for the spanning cluster, the conducting backbone, and the cutting bonds of the system. In all three cases, we observe a clear transition from ordinary to (nonlocal) explosive percolation exponents.
Departamento de Física, Universidade Federal do Ceará, 60451-970 Fortaleza, Ceará, Brazil. saulo@fisica.ufc.br
Saulo D S
SD
Reis
André A
AA
Moreira
José S
JS
Andrade
D000465:Algorithms; D003102:Colloids / Q000737:chemistry*; D003198:Computer Simulation; D008956:Models, Chemical; D008958:Models, Molecular
D003102:Colloids