On one-dimensional Cluster cluster model

TL;DR

This paper studies a one-dimensional cluster model where clusters perform random walks and merge upon contact, revealing different growth behaviors and phase transitions depending on the parameter , including finite-time infinite cluster formation.

Contribution

It provides a detailed analysis of the cluster size evolution and phase transition phenomena in the one-dimensional cluster-cluster model for various  values.

Findings

Cluster size grows as t^{1/(+2)} for  > -2

Infinite cluster forms in finite time when  < -2

Convergence in distribution of the scaling limit at =0

Abstract

The Cluster-cluster model was introduced by Meakin et al in 1984. Each $x\in \mathbb{Z}^d$ starts with a cluster of size 1 with probability $p \in (0,1]$ independently. Each cluster $C$ performs a continuous-time SRW with rate $|C|^{-\alpha}$. If it attempts to move to a vertex occupied by another cluster, it does not move, and instead the two clusters connect via a new edge. Focusing on dimension $d=1$, we show that for $\alpha>-2$, at time $t$, the cluster size is of order $t^\frac{1}{\alpha + 2}$, and for $\alpha < -2$ we get an infinite cluster in finite time a.s. Additionally, for $\alpha = 0$ we show convergence in distribution of the scaling limit.