Estimates of threshold and strength of percolation clusters on square lattices with (1,)-neighborhood

Abstract

In this paper we consider statistical estimates of threshold and strength of percolation clusters on square lattices. The percolation threshold and the strength of percolation clusters ∞ for a square lattice with (1, )-neighborhood depends not only on the lattice dimension, but also on the Minkowski exponent . To estimate the strength of percolation clusters ∞ proposed a new method of averaging the relative frequencies of the target subset of lattice sites. The implementation of this method is based on the SPSL package, released under GNU GPL-3 using the free programming language R. Keywords: site percolation, square lattice, non-metric Minkowski distance, Moore neighborhood, percolation threshold, strength of infinite clusters, R programming language, SPSL package УДК 519.676 Оценки порога и мощности перколяционных кластеров на квадратных решётках с (1, )-окрестностью П.В. Москалев Воронежский государственный аграрный университет (moskalefff@gmail.com) Аннотация. В работе рассматриваются статистические оценки порога и мощности перколяционных кластеров на квадратных решётках. Порог перколяции и мощность перколяционных кластеров ∞ на квадратной решётке с (1, )-окрестностью зависят не только от размерности решётки, но от показателя Минковского . Для оценки мощности перколяционных кластеров ∞ предложен новый метод, основанный на усреднении относительных частот целевого подмножества узлов решётки. Реализация предложенного метода основана на библиотеке SPSL, выпущенной под лицензией GNU GPL-3 с использованием свободного языка программирования R. Ключевые слова: перколяция узлов, квадратная решётка, неметрическое расстояние Минковского, окрестность Мура, порог перколяции, мощность перколяционного кластера, язык программирования R, библиотека SPSL arXiv:1311.6748v2 [cond-mat.stat-mech]

Information Technology of Problems Solutions Support in a Complex System Management

“…and "What do we want?". The use of such a model for solving elementary problems is confirmed by the modern capabilities of the method, because percolation problems are now known, posed and solved on two-dimensional [4], three-dimensional [5] and even seven-world lattices [6].…”

“…An example of such automation of transition 2 is the use of percolation models or an information model of the processes [4] to calculate optimal solutions to the problem of finding solutions with respect to some parameter (time, cost, etc.) (Fig.…”

“…Automatic control of technical processes is also based on mathematical models of the latter [3,4]. These models, as a rule, are quite complex, but their main sign is that these models are most often based on only one variable: temperature, flow rate, electric current, etc.…”

On the convergence of percolation probability functions to cumulative distribution functions on square lattices with (1,0)-neighborhood

“…This package contains several functions for generating percolation clusters and their samples on 2D and 3D square lattices with different sizes, neighborhood types, occupation probabilities and starting site subsets, but with a standard uniform weighting distribution S ∼ Unif(0, 1). The basic characteristics of percolation models realized by functions from the SPSL package have been described by the author in previously published works [16,17,18] in Russian. The scope of these percolation models is limited by such stochastic phenomena that are satisfactorily described by uniformly distributed random variables with linear cumulative distribution functions.…”

“…In this case, the sample of clusters on the percolation lattice can be interpreted as a cross-section of the random process that generates these clusters. The first numerical results with statistical analysis of site percolation models on 2D and 3D square lattices with (1, d)-neighborhoods were previously published by the author in Russian [16,17,18].…”