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TMF, 2010, Volume 165, Number 2, Pages 242–256 (Mi tmf6574)  

The critical exponent of the tree lattice generating function in the Eden model

V. E. Zobov

Kirensky Institute of Physics, Siberian Branch, RAS, Krasnoyarsk, Russia

Abstract: We consider the increase in the number of trees as their size increases in the Eden growth model on simple and face-centered hypercubic lattices in different space dimensions. We propose a first-order partial differential equation for the tree generating function, which allows relating the exponent at the critical point of this function to the perimeter of the most probable tree. We estimate tree perimeters for the lattices considered. The theoretical values of the exponents agree well with the values previously obtained by computer modeling. We thus explain the closeness of the dimension dependences of the exponents of the simple and face-centered lattices and their difference from the results in the Bethe lattice approximation.

Keywords: number of lattice trees, tree perimeter, generating function, critical exponent, hypercubic lattice, Bethe lattice, Eden model

DOI: https://doi.org/10.4213/tmf6574

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English version:
Theoretical and Mathematical Physics, 2010, 165:2, 1443–1455

Bibliographic databases:

Document Type: Article
Received: 05.03.2010
Revised: 21.05.2010

Citation: V. E. Zobov, “The critical exponent of the tree lattice generating function in the Eden model”, TMF, 165:2 (2010), 242–256; Theoret. and Math. Phys., 165:2 (2010), 1443–1455

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  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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