The percolation theory is established as a useful tool in the field of pharmaceutical materials science. It is shown that percolation theory, developed for analyzing insulator–conductor transitions, can be applied to describe imperfect dc conduction in pharmaceutical microcrystalline cellulose during densification. The system, in fact, exactly reproduces the values of the percolation threshold ...

There has been a great deal of interest and activity in percolation theory since the two Saint-Flour courses, [76, 111], of 1984 and 1996 reprinted in this new edition. We present here a summary of progress since the first publications of our lecture notes. The second edition of Percolation, [77], was published in 1999 as a fairly complete, contemporary account of ‘classical’ percolation (but w...

Consider the following process, known as r-bootstrap percolation, on a graph G. Designate some initial infected set A and infect any vertex with at least r infected neighbors, continuing until no new vertices can be infected. We say A percolates if it eventually infects the entire graph. We say A is a minimal percolating set if A percolates, but no proper subset percolates. We compute the size ...

We derive percolation results in the continuum plane that lead to what appears to be a general tendency of many stochastic network models. Namely, when the selection mechanism according to which nodes are connected to each other, is sufficiently spread out, then a lower density of nodes, or on average fewer connections per node, are sufficient to obtain an unbounded connected component. We look...

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...

We describe infinite clusters which arise in nearest-neighbour percolation for so-called cocycle measures on the square lattice. These measures arise naturally in the study of random transformations. We show that infinite clusters have a very specific form and direction. In concrete situations, this leads to a quick decision whether or not a certain cocycle measure percolates. We illustrate thi...

We survey some results and applications of last percolation models of which the limiting distribution can be evaluated.

We prove that the critical probability for bond or site percolation on Z is asymptotically equal to 1/(2d) as d → ∞. If the probability of a bond (respectively site) to be occupied is γ/(2d) with γ > 1, then for the bond model the percolation probability converges as d → ∞ to the strictly positive solution y(γ) of the equation y = 1− exp(−γy). In the site model the percolation probability is as...