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

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.

THE critical exponent of the tree lattice generating function in the eden model

“…The dependence of the exponent p on d calculated by formulas (3.12) and (3.16) is depicted in Fig. 4 together with results based on data that we obtained previously using Monte Carlo simulations [10], [11]. For d = 2, we take the value ν 1 ∼ = 0.67 obtained above (see (3.10)), which results in p ≈ 5.…”

“…We studied its dependence on d for the SC and FCC lattices in [8]- [11]. The results of the simulations and the 1/d-expansion agree well.…”

“…In pursuing this program, we used Monte Carlo simulations to estimate how the total number of trees in the ensemble increases as the number n of bonds in the tree increases for simple (SC) [10] and face-centered (FCC) [11] hypercubic lattices of various dimensions d (d = 2, 3, 4, 6, 8, 10). Using the 1/d-expansion, we then derived an asymptotic formula for the tree growth parameter, which is determined as the quantity reciprocal to the coordinate of the generating function critical point [8], [11]. We observed a good agreement between the theoretical predictions and the results of the numerical experiment.…”

“…The dependences of exponents on the space dimension for the SC lattices (squares) and for the FCC lattices (triangles): , , and solid lines are the results of numerical experiments in [10], [11];…”