Explosive Percolation is Continuous, but with Unusual Finite Size Behavior

TL;DR

This paper demonstrates that explosive percolation transitions are continuous with unique finite size behaviors, characterized by non-holomorphic scaling functions and varying universality classes, challenging the notion of their first-order nature.

Contribution

The study provides evidence that explosive percolation is continuous and reveals distinct finite size scaling behaviors and universality classes among different Achlioptas processes.

Findings

Transitions are continuous despite unusual finite size scaling.

Order parameter distributions are double-humped with peaks approaching each other as system size increases.

Different models exhibit distinct critical exponents, indicating separate universality classes.

Abstract

We study four Achlioptas type processes with "explosive" percolation transitions. All transitions are clearly continuous, but their finite size scaling functions are not entire holomorphic. The distributions of the order parameter, the relative size $s_{\rm max}/N$ of the largest cluster, are double-humped. But -- in contrast to first order phase transitions -- the distance between the two peaks decreases with system size $N$ as $N^{-\eta}$ with $\eta > 0$. We find different positive values of $\beta$ (defined via $< s_{\rm max}/N > \sim (p-p_c)^\beta$ for infinite systems) for each model, showing that they are all in different universality classes. In contrast, the exponent $\Theta$ (defined such that observables are homogeneous functions of $(p-p_c)N^\Theta$) is close to -- or even equal to -- 1/2 for all models.