Geodesic
The~critical exponent of the~tree lattice generating function in the~Eden model
Teoretičeskaâ i matematičeskaâ fizika, Tome 165 (2010) no. 2, pp. 242-256

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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. 
@article{TMF_2010_165_2_a3,
     author = {V. E. Zobov},
     title = {The~critical exponent of the~tree lattice generating function in {the~Eden} model},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {242--256},
     publisher = {mathdoc},
     volume = {165},
     number = {2},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2010_165_2_a3/}
}
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V. E. Zobov. The~critical exponent of the~tree lattice generating function in the~Eden model. Teoretičeskaâ i matematičeskaâ fizika, Tome 165 (2010) no. 2, pp. 242-256. http://geodesic.mathdoc.fr/item/TMF_2010_165_2_a3/