Scaling behavior of explosive percolation on the square lattice

TL;DR

This paper investigates the unique scaling behavior of explosive percolation on a square lattice, revealing a discontinuous transition and a critical threshold around 0.5266, differing from standard percolation.

Contribution

It provides a precise estimate of the percolation threshold and demonstrates the discontinuous nature of explosive percolation on a 2D lattice, contrasting with traditional models.

Findings

Percolation threshold p_c = 0.526565 ± 0.000005

Discontinuous jump in largest cluster size at threshold

Deviation from power-law cluster-size distribution for studied sizes

Abstract

Clusters generated by the product-rule growth model of Achlioptas, D'Souza, and Spencer on a two-dimensional square lattice are shown to obey qualitatively different scaling behavior than standard (random growth) percolation. The threshold with unrestricted bond placement (allowing loops) is found precisely using several different criteria based upon both moments and wrapping probabilities, yielding p_c = 0.526565 +/- 0.000005, consistent with the recent result of Radicchi and Fortunato. The correlation-length exponent nu is found to be close to 1. The qualitative difference from regular percolation is shown dramatically in the behavior of the percolation probability P_(infinity) (size of largest cluster), the susceptibility, and of the second moment of finite clusters, where discontinuities appears at the threshold. The critical cluster-size distribution does not follow a consistent power-law for the range of system sizes we study L </- 8192) but may approach a power-law with tau > 2 for larger L.