Explosive percolation in finite dimensions

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.