By: Rui Costa
From: Universidade de Aveiro
At: Faculdade de Ciências, Ed. C1, 1.3.14
The percolation phase transitions were believed to be continuous until recently when Achlioptas et al. , in 2009, reported a discontinuous phase transition in a new so-called “explosive percolation” problem for a competition driven process. Shortly after, we have shown that this transition is actually continuous though with surprisingly tiny critical exponents of the percolation cluster size . For a wide class of representative models, we develop a strict scaling theory  of this novel transition which provides the scaling functions and critical exponents, the full set of scaling relations, and the upper critical dimension. We rigorously derived the proper order parameter and the susceptibility for explosive percolation, which differ from the percolation cluster size and the average size of the cluster of a random node, in contrast to ordinary percolation. In the present work, we describe the impact of initial conditions on these phase transitions . We analyze the evolution starting from clusters of nodes whose sizes are distributed according to a power law, and show that these initial conditions change dramatically the phase transitions. We find an initial power-law distribution producing a particularly strong effect. For this initial condition, the system is in a “critical phase” with an infinite susceptibility before the emergence of the percolation cluster. This critical phase is absent in ordinary percolation models for any initial conditions. We show that the transition from the critical phase is an infinite-order phase transition resembling the Berezinskii-Kosterlitz-
 D. Achlioptas, R. M. D’Souza, and J. Spencer, “Explosive percolation in random networks,” Science, 323, 1453 (2009).
 R. A. da Costa, S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, “Explosive percolation transition is actually continuous,” Phys. Rev. Lett., 105, 255701 (2010).
 R. A. da Costa, S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, “Solution of the explosive percolation quest: Scaling functions and critical exponents,” Phys. Rev. E, 90, 022145 (2014).
 R. A. da Costa, S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, “Solution of the explosive percolation quest. ii. infinite-order transition produced by the initial distributions of clusters,” Phys. Rev. E, 91, 032140 (2015).