Self-similar Dynamics in Percolation and Sandpile

TL;DR

This paper uncovers scale-invariant temporal patterns in percolation and sandpile models, revealing a form of dynamic self-similarity that links static critical exponents with dynamic scaling behavior.

Contribution

It introduces a novel observation of temporal self-similarity in percolation processes and establishes quantitative relations between dynamic and static critical exponents.

Findings

Identifies scale-invariant temporal patterns during percolation.

Establishes relations between dynamic scaling exponents and static critical exponents.

Observes similar self-similar dynamics in sandpile models.

Abstract

Spatial self-similarity is a hallmark of critical phenomena. We study the dynamic process of percolation, in which bonds are incrementally added to an initially empty lattice until the system becomes fully occupied. By tracking the gap -- the size increment of clusters upon bond addition -- and the corresponding merged cluster, we identify scale-invariant temporal patterns in both quantities throughout a large portion of the process. This reveals a form of temporal self-similarity that has not been reported before. We further establish quantitative relations between the dynamic scaling exponents and the conventional static critical exponents, which enable the determination of critical behavior without prior knowledge of the critical point. The same self-similar dynamics is observed in both bond and site percolation on lattices and networks, and extends to other systems such as explosive and rigidity percolation. Moreover, similar temporal scaling is found in the initial nonequilibrium evolution of the Bak-Tang-Wiesenfeld sandpile model, suggesting a dynamic critical behavior distinct from its equilibrium state. These results provide a unified framework for understanding critical dynamics and may find applications in a broad range of complex systems.