Discrete Scale Invariance in Supercritical Percolation

TL;DR

This paper uncovers macrotransition cascades in supercritical percolation, where the order parameter exhibits discrete macroscopic steps governed by discrete scale invariance, revealing new scaling laws in non self-averaging systems.

Contribution

It reports the discovery of macrotransition cascades following percolation, demonstrating discrete scale invariance in the growth of the order parameter in supercritical percolation.

Findings

Order parameter grows in discrete macroscopic steps.

Transition positions follow scaling laws from discrete scale invariance.

Ensemble measurements reveal discrete scale invariance through rescaling.

Abstract

Recently it has been demonstrated that the connectivity transition from microscopic connectivity to macroscopic connectedness, known as percolation, is generically announced by a cascade of microtransitions of the percolation order parameter [Chen et al., Phys. Rev. Lett. 112, 155701 (2014)]. Here we report the discovery of macrotransition cascades which follow percolation. The order parameter grows in discrete macroscopic steps with positions that can be randomly distributed even in the thermodynamic limit. These transition positions are, however, correlated and follow scaling laws which arise from discrete scale invariance and non self-averaging, both traditionally unrelated to percolation. We reveal the discrete scale invariance in ensemble measurements of these non self-averaging systems by rescaling of the individual realizations before averaging.