Explosive percolation in finite dimensions

Explosive percolation (EP) has received significant research attention due to its rich and anomalous phenomena near criticality. In our recent study [], 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>d</a:mi><a:mo>=</a:mo><a:mn>2</a:mn></a:mrow></a:math> 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 <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msub><b:mi mathvariant="script">T</b:mi><b:mi>L</b:mi></b:msub></b:math> deviating from the infinite-lattice criticality by a distance determined by the <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>d</d:mi></d:math>-dependent correlation-length exponent, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:msub><e:mi mathvariant="script">T</e:mi><e:mi>L</e:mi></e:msub></e:math> follows a normal (Gaussian) distribution across all dimensions, with a standard deviation proportional to <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mn>1</g:mn><g:mo>/</g:mo><g:msqrt><g:mi>V</g:mi></g:msqrt></g:mrow></g:math>, where <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mi>V</h:mi></h:math> denotes the system volume. A theoretical argument associated with the central-limit theorem is further proposed to understand the probability distribution of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:msub><i:mi mathvariant="script">T</i:mi><i:mi>L</i:mi></i:msub></i:math>. These findings offer a comprehensive understanding of critical behaviors in EPs across various dimensions, revealing a different dimension-dependence compared to standard bond percolation. Published by the American Physical Society 2024