Achlioptas processes are not always self-averaging

TL;DR

This paper investigates Achlioptas processes in percolation models and demonstrates that, unlike classical models, some of these processes do not exhibit self-averaging, with fluctuations persisting even in large systems.

Contribution

The study reveals that self-averaging is not universal in Achlioptas processes, highlighting the presence of persistent fluctuations in certain models.

Findings

Some Achlioptas processes lack self-averaging.

Order parameter fluctuations persist in the thermodynamic limit.

Contrasts with classical percolation models.

Abstract

We consider a class of percolation models, called Achlioptas processes, discussed in [Science 323, 1453 (2009)] and [Science 333, 322 (2011)]. For these the evolution of the order parameter (the rescaled size of the largest connected component) has been the main focus of research in recent years. We show that, in striking contrast to `classical' models, self-averaging is not a universal feature of these new percolation models: there are natural Achlioptas processes whose order parameter has random fluctuations that do not disappear in the thermodynamic limit.