"Explosive percolation" transition is actually continuous
By: Rui Costa
From: Univ. Aveiro
At: Instituto de Investigação Interdisciplinar, Anfiteatro
[2011-03-10]
Recently a discontinuous percolation transition was reported in a new “explosive percolation†problem [D. Achlioptas, R.M. D’Souza, and J. Spencer, Science 323, 1453 (2009)] in striking contrast to ordinary percolation. Further investigations on this new class of irreversible problems surprisingly revealed the presence of critical scaling behavior characteristic of a continuous phase transition, making this intriguing problem one of the most urgent issues of Statistical Physics. In general, on explosive percolation models, new connections among nodes are added accordingly with microscopic rules that always involve some local optimization. We resolve the apparent contradiction by considering a representative model which, in contrast with previous models, allows conclusive analytical treatment. In particular, the rule used by our model selects the one node, of m chosen uniformly at random, belonging to the smaller cluster to connect with another randomly selected in the same way. This shows that the explosive percolation transition is actually a continuous, second order, phase transition though one with a uniquely small critical exponent of the percolation cluster size. We describe the unusual scaling properties of this transition and find its critical exponents and dimensions.