Explosive percolation on scale-free multifractal weighted planar stochastic lattice

TL;DR

This study investigates explosive bond percolation on a scale-free multifractal lattice, determining critical points and exponents, and finds that it shares scaling relations with classical percolation but exhibits a notably small beta exponent.

Contribution

First numerical determination of critical points and exponents for explosive percolation on a complex lattice, revealing similarities with classical percolation and highlighting unique exponent behavior.

Findings

Critical point $p_c$ identified numerically.

Critical exponents $eta, au, u, ext{and } ext{d}_f$ obtained.

Exponents obey classical scaling relations, with a notably small $eta$.

Abstract

In this article, we investigate explosive bond percolation (EBP) with product rule, formally known as Achlioptas process, on a scale-free multifractal weighted planar stochastic lattice (WPSL). One of the key features of the EBP transition is the delay, compared to corresponding random bond percolation (RBP), in the onset of spanning cluster. However, when it happens, it happens so dramatically that initially it was believed, albeit ultimately proved wrong, that explosive percolation (EP) exhibits first order transition. In the case of EP, much efforts were devoted to resolving the issue of its order of transition and almost no effort being devoted to find critical point, critical exponents etc., to classify it into universality classes. This is in sharp contrast to the classical random percolation. We do not even know all the exponents of EP for regular planar lattice or for Erd\"{o}s-Renyi network. We first find numerically the critical point $p_c$ and then obtain all the critical exponents $\beta, \gamma, \nu$ as well as the Fisher exponent $\tau$ and the fractal dimension $d_f$ of the spanning cluster. We also compare our results for EBP with those of the RBP and find that all the exponents of EBP obeys the same scaling relations as do the RBP. Our findings suggests that EBP is no special except the fact that the exponent $\beta$ is unusually small compared to that of RBP.