Explosive site percolation with a product rule

TL;DR

This paper investigates explosive site percolation on a 2D lattice using a product rule, revealing a discontinuous transition characterized by power-law cluster distributions and hysteresis, differing from sum rule percolation.

Contribution

It demonstrates that the product rule induces a discontinuous percolation transition in 2D, contrasting with other rules and previous continuous models.

Findings

Cluster size distribution follows a power-law with a hump near the threshold.

Transition is discontinuous, confirmed by hysteresis loop.

Transition behavior differs from sum rule percolation.

Abstract

We study the site percolation under Achlioptas process (AP) with a product rule in a $2-dimensional$ (2D) square lattice. From the measurement of the cluster size distribution, $P_s$, we find that $P_s$ has a very robust power-law regime followed by a stable hump near the transition threshold. Based on the careful analysis on the $P_s$ distribution, we show that the transition should be discontinuous. The existence of the hysteresis loop in order parameter also verifies that the transition is discontinuous in 2D. Moreover we also show that the transition nature from the product rule is not the same as that from a sum rule in 2D.