Phase Transition in the Aldous-Shields Model of Growing Trees
<|reference_start|>Phase Transition in the Aldous-Shields Model of Growing Trees: 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}.<|reference_end|>
@article{dean2005phase,
    title={Phase Transition in the Aldous-Shields Model of Growing Trees},
    author={David S. Dean and Satya N. Majumdar},
    journal={J. Stat. Phys 124, 1351 (2006).},
    year={2005},
    doi={10.1007/s10955-006-9193-9},
    archivePrefix={arXiv},
    eprint={cond-mat/0510429},
    primaryClass={cond-mat.stat-mech cs.DS math.PR}
}