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

We describe the effect of power-law initial distributions of clusters on ordinary percolation and its generalizations, specifically, models of explosive percolation processes based on local optimization. These aggregation processes were shown to exhibit continuous phase transitions if the evolution starts from a set of disconnected nodes. Since the critical exponents of the order parameter in explosive percolation transitions turned out to be very small, these transitions were first believed to be discontinuous. In this article we analyze the evolution starting from clusters of nodes whose sizes are distributed according to a power law. We show that these initial distributions change dramatically the position and order of the phase transitions in these problems. We find a particular initial power-law distribution producing a peculiar effect on explosive percolation, namely, before the emergence of the percolation cluster, the system is in a “critical phase” with an infinite generalized susceptibility. This critical phase is absent in ordinary percolation models with any power-law initial conditions. The transition from the critical phase is an infinite-order phase transition, which resembles the scenario of the Berezinskii-Kosterlitz-Thouless phase transition. We obtain the critical singularity of susceptibility at this peculiar infinite-order transition in explosive percolation. It turns out that susceptibility in this situation does not obey the Curie-Weiss law.

Figure 1

Scaling function $f\left(x\right)$ calculated using the first 1000 terms in the series, (23), for $a_0=1/\sqrt{2\pi }$ and different values of $\tilde{\tau }$. Inset: $f\left(x\right)$ is shown as a function of $x^{\sigma }\equiv s^{3-\tilde{\tau }}t$, where $\sigma =3-\tilde{\tau }$. In this representation the function $f$ is analytic at the origin, where its derivative at the origin changes signs when $\tilde{\tau }$ crosses $5/2$.

Figure 2

Numerical solution of the evolution equation, (3), for $s\le {10}^5$ with the initial condition $P(s,0)=a_0{s}^{-2}$, where $a_0=\zeta {\left(2\right)}^{-1}$. Solid lines: $P(s,t)s^2$ vs $tlns$ curves for the cluster sizes $s$ indicated by the numbers in the plot. Dashed line: Scaling function from Eq. (45), $f(tlns)=a_0/(1-2a_{0}tlns)$. Inset: Highlight of the small-$tlns$ region, in which the curves depicted here approach $f(tlns)$.

Figure 3

Numerical solution of the evolution equation, (51), for $m=2$ and $s\le {10}^5$ with the initial condition $P(s,0)=a_0{s}^{-4/3}$, where $a_0=\zeta {(4/3)}^{-1}$. Solid lines: $P(s,t)s^{4/3}$ vs $t$ curves for the cluster sizes $s$ indicated by the numbers in the plot. Dashed line: The function $f\left(t\right)$ in Eq. (77).

Figure 4

Logarithmic contribution to the asymptotics of $P(s,t_c)$ for (a) $m=2$ and (b) $m=3$. Solid lines: $P(s,t_c)s^{2m/(2m-1)}$ for $s$ from 1 to ${10}^5$ found numerically from the evolution equations with the initial condition $P(s,0)=a_0{s}^{-2m/(2m-1)}$, where $a_0=\zeta {[2m/(2m-1)]}^{-1}$. Dashed lines: Straight lines presented for reference.

Figure 5

Critical behavior of the susceptibility $\chi$ for (a) $m=1$, (b) $m=2$, and (c) $m=3$. Solid lines: Numerical solution of ${10}^5$ evolution equations with the initial condition $P(s,0)=\zeta {[2m/(2m-1)]}^{-1}{s}^{-2m/(2m-1)}$. Dashed lines: (a) power law in Eq. (50); (b), (c) power law in Eq. (89) assuming $\mu =1/(2m-2)$, and $C=0.024$ and $0.0088$ for $m=2$ and 3, respectively.