Multiple and Asymmetric Scalings in Explosive Percolation

TL;DR

This paper investigates explosive percolation via Achlioptas processes across multiple dimensions and graphs, revealing multiple scaling windows and fractal structures that clarify previous anomalies and refine critical exponents.

Contribution

It uncovers multiple and asymmetric scalings in explosive percolation, providing a new understanding of the phenomena and precise critical exponents through an event-based ensemble approach.

Findings

Identification of two scaling windows in Achlioptas processes

Discovery of multiple fractal structures in percolation

High-precision critical exponents determined

Abstract

Explosive percolation in the Achlioptas process has recently attracted much research attention. From extensive simulations in an event-based ensemble, we find that, in dimensions from $2$ to $6$ and on random graphs, the Achlioptas processes all have two scaling windows and multiple fractal structures. The mixing of these multiple scalings successfully explains the previously observed anomalous phenomena in the conventional ensemble, and, moreover, correct critical exponents are now determined with a high precision by the event-based method. The multiple scalings and the ensemble inequivalence may bring new insights for other statistical systems.