Continuous Percolation Phase Transitions of Two-dimensional Lattice Networks under a Generalized Achlioptas Process

TL;DR

This study explores how two-dimensional lattice networks undergo continuous percolation phase transitions under a generalized Achlioptas process, revealing that the critical behavior depends on the probability parameter and varies with the process.

Contribution

It introduces a generalized Achlioptas process for 2D lattices and demonstrates that the critical exponents depend on the probability parameter, extending understanding of universality classes.

Findings

Percolation transitions are continuous for all tested probabilities.

Critical exponents vary with the probability parameter p.

Universality class depends on the probability parameter p.

Abstract

The percolation phase transitions of two-dimensional lattice networks under a generalized Achlioptas process (GAP) are investigated. During the GAP, two edges are chosen randomly from the lattice and the edge with minimum product of the two connecting cluster sizes is taken as the next occupied bond with a probability $p$. At $p=0.5$, the GAP becomes the random growth model and leads to the minority product rule at $p=1$. Using the finite-size scaling analysis, we find that the percolation phase transitions of these systems with $0.5 \le p \le 1$ are always continuous and their critical exponents depend on $p$. Therefore, the universality class of the critical phenomena in two-dimensional lattice networks under the GAP is related to the probability parameter $p$ in addition.