Finite-size scaling theory for explosive percolation transitions

TL;DR

This paper develops a finite-size scaling theory specifically for explosive percolation transitions, which are discontinuous, providing a new framework to analyze their critical behavior.

Contribution

The authors introduce a finite-size scaling theory for explosive percolation, deriving a scaling function based on the divergence of the order parameter's derivative at the critical point.

Findings

Scaling function accurately collapses simulation data across system sizes

Derivative of the order parameter diverges with system size in a power-law manner

Susceptibility follows the same scaling form as the order parameter

Abstract

The finite-size scaling (FSS) theory for continuous phase transitions has been useful in determining the critical behavior from the size dependent behaviors of thermodynamic quantities. When the phase transition is discontinuous, however, FSS approach has not been well established yet. Here, we develop a FSS theory for the explosive percolation transition arising in the Erd\H{o}s and R\'enyi model under the Achlioptas process. A scaling function is derived based on the observed fact that the derivative of the curve of the order parameter at the critical point $t_c$ diverges with system size in a power-law manner, which is different from the conventional one based on the divergence of the correlation length at $t_c$. We show that the susceptibility is also described in the same scaling form. Numerical simulation data for different system sizes are well collapsed on the respective scaling functions.