Solution of the explosive percolation quest: scaling functions and critical exponents.

Abstract

Percolation refers to the emergence of a giant connected cluster in a disordered system when the number of connections between nodes exceeds a critical value. The percolation phase transitions were believed to be continuous until recently when, in a new so-called "explosive percolation" problem for a competition-driven process, a discontinuous phase transition was reported. The analysis of evolution equations for this process showed, however, that this transition is actually continuous, though with surprisingly tiny critical exponents. For a wide class of representative models, we develop a strict scaling theory of this exotic transition which provides the full set of scaling functions and critical exponents. This theory indicates the relevant order parameter and susceptibility for the problem and explains the continuous nature of this transition and its unusual properties.