Non-Local Product Rules for Percolation

TL;DR

This paper explores how introducing non-locality into the product rule percolation model affects its critical behavior, revealing a continuous spectrum of scaling exponents between ordinary and explosive percolation.

Contribution

It generalizes the product rule by incorporating non-local bond selection, demonstrating its impact on the universality class and critical exponents in percolation.

Findings

Power-law decay of bond selection influences critical exponents.

Continuous variation from ordinary to explosive percolation exponents.

Non-locality alters finite-size scaling properties.

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.