In 1982, Tsallis derived a formula which proposed an exact value of 0.522372078 . . . for the bond percolation threshold of the kagome lattice. We use the substitution method, which is based on stochastic ordering, to compare the probability distribution of connections in the homogeneous bond percolation model on the kagome lattice to those of an exactly-solved inhomogeneous bond percolation mo...

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...