Cluster expansion in cite percolation problem on cubic lattice

The cluster expansion of the Bernoulli random field percolation probability of the cubic lattice has been built. On its basis, it has been obtained the upper guaranteed estimate of the percolation threshold and corresponding accuracy estimates are proposed when some approximations are built. Построе...

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Veröffentlicht in:	Вопросы атомной науки и техники
Datum:	2012
Hauptverfasser:	Antonova, E.S., Virchenko, Yu.P.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Belgorod State University 2012
Schlagworte:	Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases
Online Zugang:	https://nasplib.isofts.kiev.ua/handle/123456789/107533
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Zitieren:	Cluster expansion in cite percolation problem on cubic lattice / E.S. Antonova, Yu.P. Virchenko // Вопросы атомной науки и техники. — 2012. — № 1. — С. 321-323. — Бібліогр.: 3 назв. — англ.

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spelling	Antonova, E.S. Virchenko, Yu.P. 2016-10-21T18:25:49Z 2016-10-21T18:25:49Z 2012 Cluster expansion in cite percolation problem on cubic lattice / E.S. Antonova, Yu.P. Virchenko // Вопросы атомной науки и техники. — 2012. — № 1. — С. 321-323. — Бібліогр.: 3 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/107533 PACS: 02.50.Cw The cluster expansion of the Bernoulli random field percolation probability of the cubic lattice has been built. On its basis, it has been obtained the upper guaranteed estimate of the percolation threshold and corresponding accuracy estimates are proposed when some approximations are built. Построено кластерное разложение для вероятности перколяции бернуллиевского случайного поля на кубической решётке. На его основе получена верхняя гарантированная оценка для порога перколяции и даны оценки точности получаемых при использовании приближений. Побудовано кластерний розклад для імовірності перколяцiї бернулiєвського випадкового поля на кубічній решітці. На його основі одержана верхня гарантована оцінка для порогу перколяцiї i подані оцінки точності наближень, якi застосовуються. en Belgorod State University Вопросы атомной науки и техники Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases Cluster expansion in cite percolation problem on cubic lattice Кластерное разложение в перколяционной задаче узлов на кубической решётке Кластерний розклад у перколяцiйнiй проблемi вузлiв на кубiчнiй решітці Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Cluster expansion in cite percolation problem on cubic lattice
spellingShingle	Cluster expansion in cite percolation problem on cubic lattice Antonova, E.S. Virchenko, Yu.P. Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases
title_short	Cluster expansion in cite percolation problem on cubic lattice
title_full	Cluster expansion in cite percolation problem on cubic lattice
title_fullStr	Cluster expansion in cite percolation problem on cubic lattice
title_full_unstemmed	Cluster expansion in cite percolation problem on cubic lattice
title_sort	cluster expansion in cite percolation problem on cubic lattice
author	Antonova, E.S. Virchenko, Yu.P.
author_facet	Antonova, E.S. Virchenko, Yu.P.
topic	Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases
topic_facet	Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases
publishDate	2012
language	English
container_title	Вопросы атомной науки и техники
publisher	Belgorod State University
format	Article
title_alt	Кластерное разложение в перколяционной задаче узлов на кубической решётке Кластерний розклад у перколяцiйнiй проблемi вузлiв на кубiчнiй решітці
description	The cluster expansion of the Bernoulli random field percolation probability of the cubic lattice has been built. On its basis, it has been obtained the upper guaranteed estimate of the percolation threshold and corresponding accuracy estimates are proposed when some approximations are built. Построено кластерное разложение для вероятности перколяции бернуллиевского случайного поля на кубической решётке. На его основе получена верхняя гарантированная оценка для порога перколяции и даны оценки точности получаемых при использовании приближений. Побудовано кластерний розклад для імовірності перколяцiї бернулiєвського випадкового поля на кубічній решітці. На його основі одержана верхня гарантована оцінка для порогу перколяцiї i подані оцінки точності наближень, якi застосовуються.
issn	1562-6016
url	https://nasplib.isofts.kiev.ua/handle/123456789/107533
citation_txt	Cluster expansion in cite percolation problem on cubic lattice / E.S. Antonova, Yu.P. Virchenko // Вопросы атомной науки и техники. — 2012. — № 1. — С. 321-323. — Бібліогр.: 3 назв. — англ.
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first_indexed	2025-11-26T09:48:30Z
last_indexed	2025-11-26T09:48:30Z
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fulltext	CLUSTER EXPANSION IN CITE PERCOLATION PROBLEM ON CUBIC LATTICE E.S. Antonova and Yu.P. Virchenko ∗ Belgorod State University, 308015, Belgorod, Russia (Received November 12, 2011) The cluster expansion of the Bernoulli random field percolation probability of the cubic lattice has been built. On its basis, it has been obtained the upper guaranteed estimate of the percolation threshold and corresponding accuracy estimates are proposed when some approximations are built. PACS: 02.50.Cw 1. INTRODUCTION We consider the set Z3 consisting of elements which are named vertexes. Let ϕ be the binary symmetric relation defined on this set. It is named the adjacency and it is such that for each vertex x = 〈n1, n2, n3〉, all vertexes y ∈ {x±ek; k = 1, 2, 3} where e1 = 〈1, 0, 0〉, e2 = 〈0, 1, 0〉, e3 = 〈0, 0, 1〉 are adjacent to x. Such adjacent relation transforms Z3 to the infinite peri- odic graph [1] which is named the cubic lattice and, further, we denote it by the same symbol Z3. Let {c̃(x);x ∈ Z3} be the random uniform Bernoulli field with values {0, 1} on the graph Z3. It is completely defined by the probability c = Pr{c̃(x) = 1} named the concentration. For each random realization c̃(x), x ∈ Z3, there exists the set W̃ = {x : c̃(x) = 1} one-to-one compared to it. It is named the configuration W̃ . The probability dis- tribution of random configurations W̃ is completely defined by the random field {c̃(x);x ∈ Z3}. The sequence γ = 〈x0,x1, ...,xn〉 consisting of vertexes belonging to the configuration W̃ and such that xi−1ϕxi, i = 1 ÷ n and xj �= xk at j �= k is named the nonintersecting path γ on the W̃ having the length n. Paths define the equivalence relation on W̃ which is named the connection. If there is the infinite nonintersecting path γ(x) on W̃ having infinite length and the initial vertex x then one may say that there exists the percolation on W̃ from the vertex x. If the percolation prob- ability P (c,x) = Pr{∃(γ(x) ⊂ W̃ )} > 0 then one may say that the Bernoulli random field has the per- colation from the vertex x at the concentration c. For the cubic lattice, due to its uniformity, the per- colation probability does not depend on the vertex x, P (c,x) ≡ P (c). Therefore, there exists the value c∗ = inf{c : P (c) > 0} named the percolation thresh- old. 2. CLUSTER EXPANSION Since the connection relation is defined on configura- tions by means of paths settled in them, it decom- poses each configuration on connected nonintersect- ing components as each equivalence relation is done. They are named clusters. If the cluster belonging to the configuration W̃ contains the vertex x, it is de- noted by W̃ (x). If the cluster W̃ (x) is infinite, and only in this case, there exists the percolation on the corresponding configuration. Let Γ be the class of all finite clusters containing the vertex 0. Consequently, the probability Pr{c̃(0) = 1} = c of the event when the zero vertex is fulled up is summarized of the fol- lowing probabilities. Firstly, it is all probabilities of events when the chosen vertex is contained in one of clusters from the class Γ, and, secondly, it is the prob- ability of the event when this vertex is contained in the infinite cluster. Consequently, it is fulfilled the so-called cluster expansion [2] c = P (c) + ∑ W∈Γ P (W ), (1) here P (W ) = Pr{W ⊂ W̃} is the probability of the fact that the cluster W is contained in the configura- tion W̃ . The cluster expansion is converged at each concentration c according to its building. Therefore, it may be the source of approximations of the perco- lation threshold in that case when one may find some upper estimates of remainders corresponding to ini- tial terms of series (1). In such a case one may ob- tain the upper estimate of the percolation threshold c∗. Besides, generally speaking, more accurate upper estimates of the percolation threshold are obtained by more accurate estimates of series remainders. ∗Corresponding author E-mail address: virch@bsu.edu.ru PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1. Series: Nuclear Physics Investigations (57), p. 321-323. 321 Upper estimates of the series (1) are built on the basis of the concept of external border ∂+W [1] of each finite cluster W . The border ∂W of the finite cluster W is the set {x �∈ W : ∃(yϕx : y ∈ W )}. The external border ∂+W of the finite cluster W is the set {x �∈ W : ∃(γ(x) ⊂ Z3 : γ(x)∩(∂W ∪W \{x}) = ∅)}. It is fulfilled the elementary estimate P (W ) < (1 − c)\|∂+W \| . (2) Obtaining of upper estimates of series (1) remainders is based on the using of this inequality. However, in the connection with the inequality (2), it is plausible to do them by the following way: to introduce the class Δ of all sets such that each of them may be ex- ternal border of anything finite cluster containing the vertex 0. In this case, the upper estimate of the sum of probabilities P (W ) on all clusters W correspond- ing to the same external border S ∈ Δ takes place. This estimate is analogous to (2), P [S] ≡ ∑ W∈Δ : ∂+W=S P (W ) < (1 − c)\|S\| . (3) Then the upper estimate takes place ∑ S∈Δ: \|S\|≥n P [S] < ∞∑ l=n (1 − c)lNl, (4) where Nl is the number of sets belonging to the class Δ and having the “area” \|S\| being less than l. Estimation of the series convergence interval and, therefore, obtaining of the upper estimate of percola- tion threshold are reduced to calculation of remainder estimates of the last series in (4). 3. UPPER ESTIMATE OF NUMBER Nl Thus, the important source of upper estimates of se- ries remainders are produced by possibly more accu- rate upper estimates of main term of the asymptotic ln Nl when l → ∞. In problems of discrete percola- tion theory on plane lattices, the upper estimates ob- taining of the number Nl is based on the Kesten the- orem which consists of the assertion: for each plane lattice, the class Δ consists of simple closed contours on the conjugate lattice which surround the vertex 0. Therefore, we obtain the reduction of the problem under consideration to the estimation of the num- ber of all pointed out contours having the length l. We have proved geometrical theorem analogous to Kesten’s one when the lattice is cubic. Theorem 1. For cubic lattice, the class Δ con- sists of all sets on it such that they are connected on the conjugate lattice and may be imbedded by home- omorphic way without edge intersection on closed ori- ented surface in R3. Each set of the class Δ decom- poses the surface on foursided faces when this imbed- ding is done. The proved theorem permits to construct the enu- meration algorithm of the class Δ and, therefore, to find upper estimates of the number Nl. Ii is proved the following assertion. Theorem 2. For cubic lattice Z3, it takes place the inequality Nl < l − 2 4 5l−1 . (5) 4. UPPER ESTIMATE OF PERCOLATION THRESHOLD Upper estimates of percolation threshold are obtained on the basis of lower estimates of those concentrations Pr{c̃(0) = 1} = c for which the series ∑ l (1−c)lNl [3] is converged. In a result, as a consequence from the inequality (5), we have gone to the following asser- tion. Theorem 3. The percolation threshold of uni- form Bernoulli field on cubic lattice satisfies the in- equality c∗ < 0, 8. At last, the estimate (5) gives the possibility to find the accuracy of approximation of the percolation probability. In particular, in the case of restriction by one series term P (W ) = c(1 − c)6 corresponding to the cluster {0}, we obtain that the accuracy of such a probability approximation is given by the estimate c(1 − (1 − c)6) − P (c) ≤ ∞∑ l=10, l– even (1 − c)lNl ≤ ≤ 1 20 ∞∑ l=10, l– even (l − 2)[5(1 − c)]l, when c > 0, 8. 5. CONCLUSIONS We have found the guaranteed estimates of perco- lation threshold in the cite percolation problem on three-dimensional lattice. Up to now rigorous math- ematical results in discrete percolation theory con- nected with concrete evaluation or accurate estima- tion percolation characteristics are known only for two-dimensional lattices. References 1. H. Kesten. Percolation Theory for Mathemati- cians. Boston: Birkhauser, 1982. 2. Yu.P. Virchenko, Yu.A. Tolmacheva. Method of Sequential Approximative Estimates in Descrete Percolation Theory // Studies in Mathemati- cal Physics Research / Ed. Charles V. Benton, New York: Nova Science Publishers, Inc., 2004, p. 155-175. 3. M.V. Menshikov, S.A. Molchanov, A.F. Sido- renko. Percolation theory and its applications // Itogi nauki i tekhniki. Ser. “Teor. ver., mat. stat. i teor. kiber”. Moscow: VINITI, 1986, v. 24, p. 53-110 (in Russian). 322 �� ! �� "�� # � � �� ! �� $ �� %�� &'&

Cluster expansion in cite percolation problem on cubic lattice

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