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.PACS numbers: 64.60.ah, 05.40.-a, 64.60.F-The percolation phase transition is one of the central issues for disordered systems [1][2][3][4]. Phase transitions in classical percolation problems are very well known to be continuous, that is, the relative size of the percolation cluster S, which is the order parameter for these models, emerges continuously, without a jump at the percolation threshold. As a continuous phase transition, the ordinary percolation transition is characterized by a power-law distribution of cluster sizes at the percolation threshold and a set of standard scaling properties and relations.This common understanding of percolation was shaken by work [5] that reported a discontinuous percolation phase transition in models whose evolution was driven by local optimization algorithms. Based on a computer experiment for a 512 , 000 node system [5], it was concluded that the percolation transition for these irreversible processes is discontinuous, and that is why this kind of percolation was termed "explosive percolation". This conclusion was supported by a number of simulations of models of this kind [6][7][8][9][10][11][12][13][14][15]. Surprisingly, these and other studies, in addition, reported power-law cluster size distributions at the critical point and scaling features below and above t c (see ), unexpected for discontinuous transitions.We resolved this contradiction by showing that the explosive percolation transition is actually continuous though with a uniquely small critical exponent β of the percolation cluster size [19]. We obtained this result by analyzing evolution equations for this process in the infinite system size limit. Thanks to the smallness of the exponent β, the continuous transition looks so "sharp" that it is virtually impossible to distinguish it from a discontinuous one in computer experiments even for very large systems [19]. More recently, the fact that this transition is continuous was also supported by mathematicians [23]. Nonetheless, in the physics sense, the quest * sdorogov@ua.pt FIG. 1. Illustration of rules in the model of explosive percolation. At each step, two sets of m nodes are chosen at ...

Solution of the explosive percolation quest. II. Infinite-order transition produced by the initial distributions of clusters

“…In particular, for ordinary percolation, ifτ = 3, the percolation cluster emerges as S ∼ exp(−const/t), where S is the relative size of this cluster. In contrast, for explosive percolation, we find that there exists a value τ < 2 + 1/(2m − 1)τ = 2 + 1/(2m − 1)τ > 2 + 1/(2m − 1) µ = 1 λ(2m − 1) − 1 τ − 2 ∼ β ∼ e −1.43m [21] χ ∼ t The initial distribution P (s, t = 0) ∼ s 1−τ ,τ > 2, S is the relative size of the percolation cluster, and χ is the susceptibility. Atτ = 2 + 1/(2m − 1), we show only the most singular factor of S.…”

“…Forτ > 2 + 1/(2m− 1) the transition occurs at t c > 0, with the critical exponents and scaling functions calculated in [15,21]. Ifτ < 2 + 1/(2m − 1) the size of the percolation cluster follows the power-law S ∼ = Bt β , with β and B given by Eqs.…”

“…Specifically, we focused on explosive percolation models, but our approach could be applied to a much wider range of generalized percolation models that can be reduced to various aggregation processes. In particular, we considered initial cluster size distributions, for which the percolation phase transition turned out to be remarkably different from that for the evolution started from isolated nodes [14,15,21]. Our results are summarized in Table I.…”

“…We consider the following explosive percolation model [14,15,21]. The evolution starts from a given distribution of clusters.…”

“…We obtained the full set of relevant critical exponents and scaling functions in the typical situation, in which the evolution starts from isolated nodes or clusters with sufficiently rapidly decaying size distribution. In our papers [14,15,21] we employ the following model of explosive percolation, at which the number of nodes N is fixed and links are added one by one. At each step we choose two sets of m random nodes, from each set we select the node that is in the smallest of m clusters, and then we add a new link between these two nodes.…”