Phase Transition in the Aldous-Shields Model of Growing Trees

Details Phase Transition in the Aldous-Shields Model of Growing Trees Phase Transition in the Aldous-Shields Model of Growing Trees

2005-10-17

arxiv.org/abs/cond-mat/0510429v2

J. Stat. Phys 124, 1351 (2006).

Physics

Condensed Matter

Statistical Mechanics

Latex 17 pages, 6 figures

Scientific paper

10.1007/s10955-006-9193-9

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

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