Site-bond percolation is a natural generalization of pure site percolation and pure bond percolation. The generalization allows both sites and bonds to be randomly occupied, with probabilities ps and pb respectively, in the case of random percolation. In particular, if pb = ps, site-bond percolation reduces to a pure site or pure bond percolation on a decorated lattice. It is also possible to i...

A hybrid model for soils, which combines percolation theory and finite element method is presented. The internal soil structure is modelled via the finite element method, and percolation networks are used for analyzing its mechanical behaviour. Through a microscopic characterization of elastic properties of soil grains, the model is generated. The effective percolation threshold obtained is low...

In this paper, we discuss the basic elements of percolation theory, beginning with the physical motivation for the abstraction and the essentials of the bond model on the LD square lattice. The critical phenomenon is introduced, a fundamental concept in percolation theory. We then describe the general use of percolation theory to analyze physical phenomena. We examine one such application in de...

We discuss inequalities and applications for percolation and randomcluster models. The relevant areas of methodology concern the following two types of inequality: inequalities involving the probability of a general increasing event, and certain differential inequalities involving the percolation probability. We summarise three areas of application of such inequalities, namely strict inequality...

Given ω ≥ 1, let Z2(ω) be the graph with vertex set Z 2 in which two vertices are joined if they agree in one coordinate and differ by at most ω in the other. (Thus Z2(1) is precisely Z .) Let pc(ω) be the critical probability for site percolation in Z2(ω). Extending recent results of Frieze, Kleinberg, Ravi and Debany, we show that limω→∞ ωpc(ω) = log(3/2). We also prove analogues of this resu...

Percolation is one of the simplest and nicest models in probability theory/statistical mechanics which exhibits critical phenomena. Dynamical percolation is a model where a simple time dynamics is added to the (ordinary) percolation model. This dynamical model exhibits very interesting behavior. Our goal in this survey is to give an overview of the work in dynamical percolation that has been do...

Using a maximum entropy principle to assign a statistical weight to any graph, we introduce a model of random graphs with arbitrary degree distribution in the framework of standard statistical mechanics. We compute the free energy and the distribution of connected components. We determine the size of the percolation cluster above the percolation threshold. The conditional degree distribution on...

As the aspect ratio of a unit (here, a stiff high-aspect-ratio rod) increases, the tendency increases for ensembles of these units to form hierarchical structures on scales quite removed from that of the single unit (particles). This is due in large part to geometrical packing considerations (based on excluded volume analysis) arising from the non-spherical shape. Nematic ordering of rod ensemb...

A first-order percolation transition, called explosive percolation, was recently discovered in evolution networks with random edge selection under a certain restriction. For many real world networks, the mechanism of preferential attachment plays a significant role in the formation of heterogeneous structures, but the network percolation in evolution process with preferential attachment has not...