Achlioptas processes are not always self-averaging

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.

Figure 1

Estimated value of ${\sigma }_n/{\mu }_n$ for ${\rho }_n\left(1\right)$ as a function of $n$ (each based on ${10}^4$ runs) using the NG, mER, and ER rules and the product rule (PR) [4]. Self-averaging (ER, PR) does not correlate with being “explosive” (NG, PR).

Figure 2

Simulation of the order parameter ${\rho }_n\left(1\right)=L_1\left(n\right)/n$ as a function of $n$ for the NG and mER rules: average and standard deviation (${10}^4$ runs for each $n$).

Figure 3

Simulation of the order parameter ${\rho }_n\left(t\right)=L_1\left(tn\right)/n$ as a function of $t$ for the NG and mER rules with $n={10}^{11}$: five sample runs together with min, max, and average of ${10}^3$ runs.