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

TL;DR

This paper investigates how power-law initial cluster distributions influence explosive percolation, revealing an infinite-order phase transition and a critical phase absent in ordinary percolation, with implications for understanding complex network behaviors.

Contribution

It demonstrates that specific initial power-law distributions induce an infinite-order transition and a critical phase in explosive percolation models, differing from traditional models.

Findings

Initial distributions dramatically alter phase transition behavior.

Identification of a critical phase with infinite susceptibility.

Discovery of an infinite-order transition resembling BKT transition.

Abstract

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 the susceptibility in this situation does not obey the Curie-Weiss law.