Explosive percolations on the Bethe Lattice

TL;DR

This paper introduces a new self-consistent simulation method for analyzing percolation on the Bethe lattice, revealing that Achlioptas process models exhibit continuous phase transitions.

Contribution

The authors develop a novel self-consistent simulation approach for arbitrary percolation on the Bethe lattice, enabling analysis of phase transition types in Achlioptas processes.

Findings

Achlioptas processes on the Bethe lattice undergo continuous transitions.

The SCS method accurately models both continuous and discontinuous percolation.

Achlioptas models show continuous transitions regardless of growth rule details.

Abstract

Based on the self-consistent equations of the order parameter $P_\infty$ and the mean cluster size $S$, we develop a novel self-consistent simulation (SCS) method for arbitrary percolation on the Bethe lattice (infinite homogeneous Cayley tree). By applying SCS to the well-known percolation models, random bond percolation and bootstrap percolation, we obtain prototype functions for continuous and discontinuous phase transitions. By comparing the key functions obtained from SCSs for the Achlioptas processes (APs) with a product rule and a sum rule to the prototype functions, we show that the percolation transition of AP models on the Bethe lattice is continuous regardless of details of growth rules.