Explosive percolation in finite dimensions

TL;DR

This study investigates explosive percolation in finite dimensions, confirming standard finite-size scaling, determining critical thresholds, and revealing different universality classes and distribution behaviors across dimensions 2 to 6.

Contribution

The paper provides a comprehensive simulation analysis of explosive percolation in finite dimensions, establishing its critical behaviors and universality classes, and introduces a theoretical explanation for the distribution of pseudocritical points.

Findings

Critical behaviors follow standard finite-size scaling theory.

Percolation thresholds and critical exponents are precisely determined.

Distribution of pseudocritical points is Gaussian with standard deviation proportional to 1/√V.

Abstract

Explosive percolation (EP) has received significant research attention due to its rich and anomalous phenomena near criticality. In our recent study [Phys. Rev. Lett. 130, 147101 (2023)], we demonstrated that the correct critical behaviors of EP in infinite dimensions (complete graph) can be accurately extracted using the event-based method, with finite-size scaling behaviors still described by the standard finite-size scaling theory. We perform an extensive simulation of EPs on hypercubic lattices ranging from dimensions $d=2$ to $6$, and find that the critical behaviors consistently obey the standard finite-size scaling theory. Consequently, we obtain a high-precision determination of the percolation thresholds and critical exponents, revealing that EPs governed by the product and sum rules belong to different universality classes. Remarkably, despite the mean of the dynamic pseudocritical point $\mathcal{T}_L$ deviating from the infinite-lattice criticality by a distance determined by the $d$-dependent correlation-length exponent, $\mathcal{T}_L$ follows a normal (Gaussian) distribution across all dimensions, with a standard deviation proportional to $1/\sqrt{V}$, where $V$ denotes the system volume. A theoretical argument associated with the central-limit theorem is further proposed to understand the probability distribution of $\mathcal{T}_L$. These findings offer a comprehensive understanding of critical behaviors in EPs across various dimensions, revealing a different dimension-dependence compared to standard bond percolation.