  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.
