Phase Transition in a Generalized Eden Growth Model on a Tree |
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Authors: | David S. Dean Satya N. Majumdar |
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Affiliation: | (1) Laboratoire de Physique Théorique, UMR CNRS 5152, IRSAMC, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 04, France;(2) Laboratoire de Physique Théorique et Modéles Statistiques, UMR CNRS 8626, Université Paris Sud, Bat 100, 91045 Orsay Cedex, France |
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Abstract: | We study analytically the late time statistics of the number of particles in an Eden growth model on a tree. 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 . While for the variance is proportional to the mean and the distribution is normal, for 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 . |
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Keywords: | Search trees Growth models Phase transitions |
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