We introduce and study a family of 2D percolation systems which are based on the bond percolation model of the triangular lattice. The system under study has local correlations, however, bonds separated by a few lattice spacings act independently of one another. By avoiding explicit use of microscopic paths, it is first established that the model possesses the typical attributes which are indic...

Bootstrap percolation is one of the simplest cellular automata. In r-bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: infected vertices of G remain infected forever and in consecutive rounds healthy vertices with at least r already infected neighbours become infected. Percolation occurs if eventually every vertex is infected. In this paper ...

Zhang found a simple, elegant argument deducing the non-existence of an infinite open cluster in certain lattice percolation models (for example, p = 1/2 bond percolation on the square lattice) from general results on the uniqueness of an infinite open cluster when it exists; this argument requires some symmetry. Here we show that a simple modification of Zhang’s argument requires only 2-fold (...

We postulate the existence of a natural Poissonian marking of the double (touching) points of SLE6 and hence of the related continuum nonsimple loop process that describes macroscopic cluster boundaries in 2D critical percolation. We explain how these marked loops should yield continuum versions of near-critical percolation, dynamical percolation, minimal spanning trees and related plane fillin...

Let Qd denote the hypercube of dimension d. Given d ≥ m, a spanning subgraph G of Qd is said to be (Qd, Qm)-saturated if it does not contain Qm as a subgraph but adding any edge of E(Qd) \E(G) creates a copy of Qm in G. Answering a question of Johnson and Pinto [27], we show that for every fixed m ≥ 2 the minimum number of edges in a (Qd, Qm)-saturated graph is Θ(2 d). We also study weak satura...

Realistic network-like systems are usually composed of multiple networks with interacting relations such as school-enterprise research and development (R&D) collaboration networks. Here we study the percolation properties of a special kind of that R&D collaboration networks, namely institute-enterprise R&D collaboration networks (IERDCNs). We introduce two actual IERDCNs to show their structura...

We show that when percolation produces infinitely many infinite clusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies that uniqueness Ž of the infinite cluster is equivalent to nondecay of connectivity a.k.a. . long-range order . We then derive applications concerning uniqueness in Kazhdan groups and in wreath products ...

We study mixed site-bond directed percolation on 2D and 3D lattices by using time-dependent simulations. Our results are compared with rigorous bounds recently obtained by Liggett and by Katori and Tsukahara. The critical fractions p c site and p c bond of sites and bonds are extremely well approximated by a relationship reported earlier for isotropic percolation, (log p c site / log p c * site...