We investigate the scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in high dimensions, or when d > 6 for sufficient spread-out percolation. We use a relatively simple coupling argument to show that this largest critical cluster is, with high probability, bounded above by a large constant times V 2/3 and below by a small consta...

We estimate locations of the regions of the percolation and of the non-percolation in the plane (λ, β): the Poisson rate – the inverse temperature, for interacted particle systems in finite dimension Euclidean spaces. Our results about the percolation and about the nonpercolation are obtained under different assumptions. The intersection of two groups of the assumptions reduces the results to t...

This work analyzes a percolation model on the diamond hierarchical lattice (DHL), where the percolation transition is retarded by the inclusion of a probability of erasing specific connected structures. It has been inspired by the recent interest on the existence of other universality classes of percolation models. The exact scale invariance and renormalization properties of DHL leads to recurr...

1 Executive Summary We have continued with a review on the bond percolation and the introduced the coupling of the bond percolation process, then defined the critical phenomenon and the percolation probability. There is a theorem about the percolation probability is a non decreasing function and we bounded it between 1/3 and 2/3 for the two dimensional case. 2 Bond Percolation 2.1 Preliminaries...

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.