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Zobov, V. E.; Popov, M. A.

doi:10.1023/a:1025603416535

Cited by 7 publications

(7 citation statements)

References 9 publications

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“…For the latter, the results agree well with the conclusions about the presence of singular points on the imaginary time axis, which follow because the spectra of these functions have exponential wings observed experimentally (see the references and discussions in [9]- [13]). the tree root, and second, branches can grow from vertices of the chain and of the loop.…”

Section: Discussionsupporting

confidence: 87%

“…For comparison, the dashed line in Fig. 2 represents the result for δx C /x B obtained without expanding the quantities N m and R m (x) for m = 3, 4, 5 in powers of 1/d in (13). This curve goes down at the dimensions d = 3 and d = 4 because the number N m decreases because of the multiplier (d − 2) m−2 , which we replace with d m−2 in the expansion.…”

Section: Discussionmentioning

confidence: 99%

“…The results in [11]- [13] for the coordination numbers Z and the numbers N m of different embeddings of the loop composed of m bonds with the vertex at the origin for these two lattices are collected in Table 1. Table 1 SC FCC…”

Section: Generating Functionsmentioning

confidence: 99%

“…In the growth models, we pass from analyzing properties of a separate cluster to analyzing properties of the whole ensemble of clusters based on studying their generating function [9]- [13]. Such an analysis is necessary for studying the analytic properties of time-spin correlation functions at high temperature because in the case of high-dimension lattices after passing to the imaginary time, these functions can be considered the generating functions of trees constructed following special rules.…”

Section: Introductionmentioning

confidence: 99%

“…Other lattices are also interesting. For example, face-centered cubic (FCC) lattices are very important when analyzing the results of actual NMR experiments [13]. The FCC lattices differ from the SC lattices in the form of the dependence of the coordination number on the space dimension and in the form of lattice figures.…”

Section: Introductionmentioning

confidence: 99%

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

Zobov

Popov²

2005

Theor Math Phys

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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.

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Section: Discussionsupporting

confidence: 87%

Section: Discussionmentioning

confidence: 99%

Section: Generating Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Zobov

Popov²

2005

Theor Math Phys

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

Zobov

2010

Theor Math Phys

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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.

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