It has been shown that the last passage time in certain symmetrized models of directed percolation can be written in terms of averages over random matrices from the classical groups U(l), Sp(2l) and O(l). We present a theory of such results based on non-intersecting lattice paths, and integration techniques familiar from the theory of random matrices. Detailed derivations of probabilities relat...

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...

We study random walks on supercritical percolation clusters on wedges in Z, and show that the infinite percolation cluster is (a.s.) transient whenever the wedge is transient. This solves a question raised by O. Häggström and E. Mossel. We also show that for convex gauge functions satisfying a mild regularity condition, the existence of a finite energy flow on Z is equivalent to the (a.s.) exis...

The dimpling in the CuO2 planes of overdoped Y1−yCayBa2Cu3 O6.96−6.98, (y = 0.02−0.2) has been measured by x-ray absorption-fine-structure spectroscopy (Y-K EXAFS). A step-like decrease around 12% Ca indicates a percolation threshold for distorted sites of 5 cells, and thus phase segregation. We conclude the charge carriers added by substitution of Y by Ca to be trapped at the Ca sites and thei...

Let 0 < a < b < ∞, and for each edge e of Zd let ωe = a or ωe = b, each with probability 1/2, independently. This induces a random metric distω on the vertices of Z d, called first passage percolation. We prove that for d > 1 the distance distω(0, v) from the origin to a vertex v, |v| > 2, has variance bounded by C |v|/ log |v|, where C = C(a, b, d) is a constant which may only depend on a, b a...