"Explosive Percolation" Transition is Actually Continuous

Abstract: The basic notion of percolation in physics assumes the emergence of a giant connected (percolation) cluster in a large disordered system when the density of connections exceeds some critical value. Until recently, the percolation phase transitions were believed to be continuous, however, in 2009, a remarkably different, discontinuous phase transition was reported in a new so-called "explosive percolation" problem. Each link in this problem is established by a specific optimization process. Here, employing strict analytical arguments and numerical calculations, we find that in fact the "explosive percolation" transition is continuous though with an uniquely small critical exponent of the percolation cluster size. These transitions provide a new class of critical phenomena in irreversible systems and processes.