We study roughening interfaces with a constant slope that become self organized critical by a rule that is similar to that of invasion percolation. The transient and critical dynamical exponents show Galilean invariance. The activity along the interface exhibits nontrivial power law correlations in both space and time. The probability distribution of the activity pattern follows an algebraic re...

The review is a brief description of the state of problems in percolation theory and their numerous applications, which are analyzed on base of interesting papers published in the last 15-20 years. At the submitted papers are studied both the cluster system of the physical body and its impact on the object in general, and adequate mathematical tools for description of critical phenomena too. Of...

We give accurate estimates for the bond percolation critical probabilities on seven Archimedean lattices, for which the critical probabilities are unknown, using an algorithm of Newman and Ziff.

Consider bond percolation on the square latticeZ where each edge is independently open with probability p. For some positive constants p0 ∈ (0, 1), 1 and 2, the following holds: if p > p0, then with probability at least 1− 1 n4 there are at least 2n logn disjoint open left-right crossings in Bn := [0, n] 2 each having length at most 2n, for all n ≥ 2. Using the proof of the above, we obtain pos...

We introduce a set of iterative equations that exactly solves the size distribution of components on small arbitrary graphs after the random removal of edges. We also demonstrate how these equations can be used to predict the distribution of the node partitions (i.e., the constrained distribution of the size of each component) in undirected graphs. Besides opening the way to the theoretical pre...

Short title: Isoperimetry on percolation clusters. Abstract: we prove that the heat kernel on the infinite Bernoulli percolation cluster in Z almost surely decays faster than t−d/2. We also derive estimates on the mixing time for the random walk confined to a finite box. Our approach is based on local isoperimetric inequalities. Some of the results of this paper were previously announced in the...

We consider a random subgraph Gp of a host graph G formed by retaining each edge of G with probability p. We address the question of determining the critical value p (as a function of G) for which a giant component emerges. Suppose G satisfies some (mild) conditions depending on its spectral gap and higher moments of its degree sequence. We define the second order average degree d̃ to be d̃ = ∑ v...

We present a generalized connectedness percolation theory reduced to a compact form for a large class of anisotropic particle mixtures with variable degrees of connectivity. Even though allowing for an infinite number of components, we derive a compact yet exact expression for the mean cluster size of connected particles. We apply our theory to rodlike particles taken as a model for carbon nano...

We study the percolation properties of embedded topological defects, Z–vortices, and the occurrence of the corresponding Nambu monopoles in the electroweak theory for a Higgs mass MH ≈ 100 GeV, near to the crossover temperature. Although there is no phase transition anymore at such strong selfcoupling, we observe that the Z–vortices still exhibit a percolation transition that has been found rec...

Spatial models for spread of an epidemic may be mapped onto bond percolation. We point out that with disorder in the strength of contacts between individuals patchiness in the spread of the epidemic is very likely, and the criterion for epidemic outbreak depends strongly on the disorder because the critical region of the corresponding percolation model is broadened. In some networks the percola...