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 TMF, 2005, Volume 144, Number 3, Pages 564–576 (Mi tmf1877)

Tree Growth Parameter in the Eden Model on Face-Centered Hypercubic Lattices

V. E. Zobova, M. A. Popovb

a L. V. Kirensky Institute of Physics, Siberian Branch of the Russian Academy of Sciences
b Krasnoyarsk State University

Abstract: In the Eden model, we investigate how the tree growth parameter depends on the space dimension $d$ for face-centered hypercubic lattices. We find the first three terms of the $1/d$-expansion for this parameter directly from the generating function without calculating the number of trees because the growth parameter is the reciprocal coordinate of the singular point of the tree generating function. The same growth parameter was calculated by computer experiment where the ratios between the numbers of trees without intersections and trees without restrictions in the dimensions 3, 4, 6, 8, and 10 were estimated by the Monte Carlo method on face-centered cubic lattices. The results of the two methods agree well. Comparing with the previously performed computer experiment for simple hypercubic lattices, we observe that the values of the singular exponents for the tree generating functions are close for two different types of lattices.

Keywords: Eden model, number of lattice trees, Monte Carlo method, growth parameter, singular points of generating function, large-dimension expansion, face-centered hypercubic lattice

DOI: https://doi.org/10.4213/tmf1877

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English version:
Theoretical and Mathematical Physics, 2005, 144:3, 1361–1371

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Revised: 01.12.2004

Citation: V. E. Zobov, M. A. Popov, “Tree Growth Parameter in the Eden Model on Face-Centered Hypercubic Lattices”, TMF, 144:3 (2005), 564–576; Theoret. and Math. Phys., 144:3 (2005), 1361–1371

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tmf1877
• https://doi.org/10.4213/tmf1877
• http://mi.mathnet.ru/eng/tmf/v144/i3/p564

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This publication is cited in the following articles:
1. V. E. Zobov, “The critical exponent of the tree lattice generating function in the Eden model”, Theoret. and Math. Phys., 165:2 (2010), 1443–1455
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