Finite-size effects for anisotropic bootstrap percolation: logarithmic corrections

In this note we analyze an anisotropic, two-dimensional bootstrap percolation model introduced by Gravner and Griffeath. We present upper and lower bounds on the finite-size effects. We discuss the similarities with the semi-oriented model introduced by Duarte.Comment: Key words: Bootstrap percolation, anisotropy, finite-size effect

Metastability threshold for anisotropic bootstrap percolation in three dimensions

In this paper we analyze several anisotropic bootstrap percolation models in three dimensions. We present the order of magnitude for the metastability threshold for a fairly general class of models. In our proofs we use an adaptation of the technique of dimensional reduction. We find that the order of the metastability threshold is generally determined by the "easiest growth direction" in the model. In contrast to the anisotropic bootstrap percolation in two dimensions, in three dimensions the order of the metatstability threshold for anisotropic bootstrap percolation can be equal to that of isotropic bootstrap percolation.Comment: 19 page

The histogram characteristics of perimeter polynomials for directed percolation

New perimeter polynomials (in dimensions d = 2 to 4) are analysed for directed site percolation. A study of these data shows that i) above pc the average perimeter-to-size- ratio varies as α = (1 — p) /p + Bs-1/d; ii) At pc its leading correction term estimates supports the prediction (from scaling) of an exponent equal to 1/Δ — 1 (with Δ the gap exponent for directed percolation); iii) At p = 0 the limiting ratio is estimated on various lattices. Fairly definitive evidence is obtained in favour of α(p = 0) = 3/4 for the square site animals and this result is used to study the second correction term which is estimated to be analytic (∼ s-2) as the first correction term (Bethe — like and∼ s-1, without any obvious dimensional dependence).Nous analysons de nouveaux polynomes de périmètre (en dimensions 2 à 4) dans le cas de la percolation de site dirigée. Une étude de ces données montre que i) au-dessus de Pc, le rapport moyen périmètre sur taille varie comme a = (1 - p)/p + Bs-1/d, ii) à pc, l'evaluation du terme de correction dominant corrobore la prédiction (qui vient des lois d'échelles) d'un exposant égal à 1/Δ - 1 (où Δ est l'exposant de gap de la percolation dirigée), iii) à p = 0, nous estimons la limite du rapport pour plusieurs réseaux. Nous obtenons des résultats assez concluants en faveur de α(p = 0) = 3/4 pour les animaux de site sur réseau carré et à partir de là estimons le second terme de correction. Nous le trouvons analytique (∼ s -2) comme le premier terme (du type de Bethe ∼ s-1, et apparemment indépendant de la dimension)

Site and bond percolation distributions : a survey of perimeters for all values of p

This paper presents a comprehensive survey of site and bond percolation distributions. Agreement with the scaling picture and universality for both types of percolation is found within the usual uncertainty limits. Asymptotic analysis of coefficients also supports existing predictions for the high density region. In addition, the non-critical region is studied for a variety of lattices. The results obtained enable a comparative analysis of the existing conflicting proposals for that region.On présente une étude de deux types de percolation pour différents réseaux et à plusieurs dimensions. On montre qu'au seuil de percolation, les lois d'échelle sont suivies et que l'universalité des exposants critiques est vérifiée. L'analyse asymptotique des coefficients confirme les hypothèses prévoyant l'évolution de petits amas dans la région critique. En outre, on étudie le régime non critique pour un certain nombre de réseaux particuliers. Les résultats ainsi obtenus permettent de comparer les modèles contradictoires

The branching of real lattice trees as dilute polymers

In this paper, the branching of real lattice trees is shown to be related to the occurrence of two different prefactor exponents. For all lattices where trivalent trees are embeddable, this exponent is estimated as being animal-like in nature. In addition, estimates for the growth parameter are given for a number of 2- and 3-dimensional lattices.On montre dans cet article que le branchement en arbres sur un réseau réel est lié à la présence des exposants de deux préfacteurs différents. Pour tous les réseaux où les arbres trivalents sont inscrits, cet exposant est estimé être de même nature que dans le cas des animaux. En outre, on donne des estimations du paramètre de croissance pour des réseaux à deux et trois dimensions

The perimeter in site directed percolation. Mean perimeter expansions

Exact expansions for susceptibility — like mean perimeter series for directed percolation and extended mean size series are analysed, on two and three dimensional lattices. The critical threshold pc estimates are refined for the triangular and simple cubic lattices. On the square next-nearest-neighbour site problem pc is estimated as pc = 0.4965 ± 0.002.Nous analysons des séries pour le périmètre moyen et la taille moyenne en percolation dirigée, séries dont nous avons obtenu des termes nouveaux. Nous obtenons des estimations plus précises pour le seuil de percolation critique pc dans le cas des réseaux triangulaires et cubiques. Dans le cas du problème de percolation de site avec seconds proches voisins sur le réseau carré, nous trouvons pc = 0,4965 ± 0,002

The Self-Organized Critical Forest-Fire Model with Trees and Bushes

We present a forest-fire model with trees and bushes. Additional to the usual nearest-neighbour fire spreading both populations can ignite each other at the same lattice site. Simulations in two dimensions show that the bushes with the higher growth rate remain critical while the self-organization of the trees is destroyed by quick bush fires which ignite the tree clusters before they can become large. A mean-field treatment leads to coupled equations for the densities at the stationary point, depending on the ratio $k$ of the growth rates

Finite-Size Effects for Some Bootstrap Percolation Models

The consequences of Schonmann's new proof that the critical threshold is unity for certain bootstrap percolation models are explored. It is shown that this proof provides an upper bound for the finite-size scaling in these systems. Comparison with data for one case demonstrates that this scaling appears to give the correct asymptotics. We show that the threshold for a finite system of size L scales as O{1/[ln(ln L)]} for the isotropic model in three dimensions where sites that fail to have at least four neighbors are culled.

Dynamic permeability of porous media by cellular automata

The step response of a porous medium with quenched disorder is investigated by cellular automata. The main conclusions of this dynamic study on samples of up to $2000\times 660$ sites were that the response times are a linear function of porosity in the free channel limit and exhibit also a linear dependence on the lattice, size, for identical disorder distributions. The porous section response is found to vary according to its length as a modified exponential with a power dependence on the time that increases monotonically with the porous length