We prove that there exist natural generalizations of the classical bootstrap percolation model on Z that have non-trivial critical probabilities, and moreover we characterize all homogeneous, local, monotone models with this property. Van Enter [28] (in the case d = r = 2) and Schonmann [25] (for all d > r > 2) proved that r-neighbour bootstrap percolation models have trivial critical probabili...

We formulate the Wolff algorithm as a site-bond percolation problem, apply it to the ferromagnetic and antiferromagnetic planar triangular spin models, and study the percolation critical behavior using Anite-size scaling. In the former case the Wold' algorithm is successful as an accelerating algorithm, whereas in the latter case it is not. We found the percolation temperatures and the cluster ...

We first introduce the percolation problems associated with the graph theoretical concepts of (k, l)-sparsity, and make contact with the physical concepts of ordinary and rigidity percolation. We then devise a renormalization transformation for (k, l)-percolation problems, and investigate its domain of validity. In particular, we show that it allows an exact solution of (k, l)-percolation probl...

We define a percolation problem on the basis of spin configurations of the two-dimensional XY model. Neighboring spins belong to the same percolation cluster if their orientations differ less than a certain threshold called the conducting angle. The percolation properties of this model are studied by means of Monte Carlo simulations and a finite-size scaling analysis. Our simulations show the e...

Electrical conductivity and permittivity of carbon black-polyvinyl chloride composites were studied over a wide frequency spectrum (dc, 1.3 GHz). Conductivity of the bulk composites increases with higher volume fraction of carbon black as expected. However, the functional dependence of the increasing conductivity with carbon black loading is different below and above the percolation threshold b...

We study percolation in the hierarchical lattice of order N where the probability of connection between two points separated by distance k is of the form ck/N , δ > −1. Since the distance is an ultrametric, there are significant differences with percolation in the Euclidean lattice. We consider three regimes: δ < 1, where percolation occurs, δ > 1, where it does not occur, and δ = 1 which is th...

The location and nature of the percolation transition in random networks is a subject of intense interest. Recently, a series of graph evolution processes have been introduced that lead to discontinuous percolation transitions where the addition of a single edge causes the size of the largest component to exhibit a significant macroscopic jump in the thermodynamic limit. These processes can hav...

We describe in detail an efficient algorithm for studying site or bond percolation on any lattice. The algorithm can measure an observable quantity in a percolation system for all values of the site or bond occupation probability from zero to one in an amount of time that scales linearly with the size of the system. We demonstrate our algorithm by using it to investigate a number of issues in p...

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...