We study the problem of coexistence in a two-type competition model governed by first-passage percolation on Zd or on the infinite cluster in Bernoulli percolation. We prove for a large class of ergodic stationary passage times that for distinct points x, y ∈ Zd , there is a strictly positive probability that {z ∈ Zd ;d(y, z) < d(x, z)} and {z ∈ Zd ;d(y, z) > d(x, z)} are both infinite sets. We...

The site percolation problem is studied on d-dimensional generalizations of the Kagomé lattice. These lattices are isotropic and have the same coordination number q as the hyper-cubic lattices in d dimensions, namely q = 2d . The site percolation thresholds are calculated numerically for d = 3, 4, 5, and 6. The scaling of these thresholds as a function of dimension d, or alternatively q, is dif...

In this paper we consider the AB-percolation model on Z1 d and Z. Let pH (Zd) be the critical probability for AB-percolation on Z. We show that pH (Zd) ;1/(2d). If the probability of a site to be in state A is g/(2d) for some fixed g.1, then the probability that AB-percolation occurs converges as d→` to the unique strictly positive solution y(g) of the equation y512exp(2gy). We also find the li...

We study the percolation problem on the Apollonian network model. The Apollonian networks display many interesting properties commonly observed in real network systems, such as small-world behavior, scale-free distribution, and a hierarchical structure. By taking advantage of the deterministic hierarchical construction of these networks, we use the real-space renormalization-group technique to ...

Motivated by the success of a k-clique percolation method for the identification of overlapping communities in large real networks, here we study the k-clique percolation problem in the Erd ́́ os–Rényi graph. When the probability p of two nodes being connected is above a certain threshold pc(k), the complete subgraphs of size k (the k-cliques) are organized into a giant cluster. By making some as...

simultaneous capillary dominated displacement of the wetting and non-wetting phases are processes of interest in many disciplines including modeling of the penetration of polluting liquids in hydrology or the secondary migration in petroleum reservoir engineering. percolation models and in particular invasion percolation is well suited to characterize the slow immiscible displacement of two flu...

Knowledge can be seen as a complex system that does not reveal clearly how it works. However, some mathematical models have proved day after day their relevance to the description of complex phenomena in fields that are apparently unrelated. This is especially the case for Percolation Theory. After a survey of the percolation mechanism, it will be seen that Percolation Theory has proved to be r...

–Accessibility percolation is a new type of percolation problem inspired by evolutionary biology. To each vertex of a graph a random number is assigned and a path through the graph is called accessible if all numbers along the path are in ascending order. For the case when the random variables are independent and identically distributed, we derive an asymptotically exact expression for the prob...

We introduce a new class of bootstrap percolation models where the local rules are of a geometric nature as opposed to simple counts of standard bootstrap percolation. Our geometric bootstrap percolation comes from rigidity theory and convex geometry. We outline two percolation models: a Poisson model and a lattice model. Our Poisson model describes how defects—holes is one of the possible inte...