Percolation with Multiple Giant Clusters

TL;DR

This paper investigates how freezing influences percolation, revealing a threshold that suppresses the transition and leads to multiple fluctuating giant clusters with a universal size distribution tail.

Contribution

It introduces a model of percolation with freezing, demonstrating the suppression of the transition and the emergence of multiple giant clusters with universal properties.

Findings

Percolation transition is suppressed above a freezing rate threshold.

Multiple giant clusters form below the threshold with fluctuating sizes.

Frozen cluster size distribution follows a universal tail F_k ~ k^{-3}.

Abstract

We study the evolution of percolation with freezing. Specifically, we consider cluster formation via two competing processes: irreversible aggregation and freezing. We find that when the freezing rate exceeds a certain threshold, the percolation transition is suppressed. Below this threshold, the system undergoes a series of percolation transitions with multiple giant clusters ("gels") formed. Giant clusters are not self-averaging as their total number and their sizes fluctuate from realization to realization. The size distribution F_k, of frozen clusters of size k, has a universal tail, F_k ~ k^{-3}. We propose freezing as a practical mechanism for controlling the gel size.