Phase Transition in the Aldous-Shields Model of Growing Trees

TL;DR

This paper analytically investigates the late-time statistics of a growing tree model, revealing a phase transition in variance behavior at a critical parameter value, with implications for understanding fluctuations in such stochastic growth processes.

Contribution

The study introduces a backward Fokker-Planck approach to analyze the Aldous-Shields model, identifying a phase transition in variance and distribution type at a critical parameter value.

Findings

Variance is proportional to mean for c > sqrt(2)

Variance is anomalously large and distribution is non-Gaussian for c < sqrt(2)

Critical point occurs at c = sqrt{m} for generalized m-branch trees

Abstract

We study analytically the late time statistics of the number of particles in a growing tree model introduced by Aldous and Shields. In this model, a cluster grows in continuous time on a binary Cayley tree, starting from the root, by absorbing new particles at the empty perimeter sites at a rate proportional to c^{-l} where c is a positive parameter and l is the distance of the perimeter site from the root. For c=1, this model corresponds to random binary search trees and for c=2 it corresponds to digital search trees in computer science. By introducing a backward Fokker-Planck approach, we calculate the mean and the variance of the number of particles at large times and show that the variance undergoes a `phase transition' at a critical value c=sqrt{2}. While for c>sqrt{2} the variance is proportional to the mean and the distribution is normal, for c<sqrt{2} the variance is anomalously large and the distribution is non-Gaussian due to the appearance of extreme fluctuations. The model is generalized to one where growth occurs on a tree with $m$ branches and, in this more general case, we show that the critical point occurs at c=sqrt{m}.