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arXiv · 1112.0557

Non-Local Product Rules for Percolation

S. D. S. Reis·A. A. Moreira·J. S. Andrade Jr
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Abstract

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 non-locality 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 continuous variation from ordinary to (non-local) explosive percolation exponents.

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S. D. S. Reis, A. A. Moreira, J. S. Andrade Jr. 2011-12-02. Non-Local Product Rules for Percolation. https://doi.org/10.1103/physreve.85.041112

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