We present high statistics data on the distribution of shortest path lengths between two near-by points on the same cluster at the percolation threshold. Our data are based on a new and very efficient algorithm. For d = 2 they clearly disprove a recent conjecture by M. Porto et al., Phys. Rev. E 58, R5205 (1998). Our data also provide upper bounds on the probability that two near-by points are ...

In this article, I give a pedagogical introduction and overview of percolation theory. Special emphasis will be put on the review of some of the most prominent of the algorithms that have been devised to study percolation numerically. At the central stage shall be the real-space renormalization group treatment of the percolation problem. As a rather novel application of this approach to percola...

We study the magnetization distribution of the Ising model on two regular fractals (a hierarchical lattice, the regular simplex) and percolation clusters at the percolation threshold in a two dimensional imbedding space. In all these cases, the only fixed point is T = 0. In the case of the two regular fractals, we show that the magnetization distribution is non trivial below T ∗ ≃ A/n, with n t...

A method for obtaining a 1/σ expansion for certain statistical models is presented, where σ+1 is the coordination number of the lattice. The method depends on being able to generate exact recursion relations for the Cayley tree. By perturbing the recursion relation to take account of the dominant loops in a hypercubic lattice for large σ, we obtain corrections of order σ−2 to the recursion rela...

This paper is an up-to-date introduction to the problem of uniqueness versus non-uniqueness of infinite clusters for percolation on Z d and, more generally, on transitive graphs. For iid percolation on Z, uniqueness of the infinite cluster is a classical result, while on certain other transitive graphs uniqueness may fail. Key properties of the graphs in this context turn out to be amenability ...

We postulate the existence of a natural Poissonian marking of the double (touching) points of SLE6 and hence of the related continuum nonsimple loop process that describes macroscopic cluster boundaries in 2D critical percolation. We explain how these marked loops should yield continuum versions of near-critical percolation, dynamical percolation, minimal spanning trees and related plane fillin...

In this paper, we jointly consider the inhomogeneity and spatial dimension in large scale wireless networks. We study the effects of topology on the throughput capacity. This problem is inherently difficult since it is complex to handle the interference caused by simultaneous transmission. To solve this problem, we, according to the inhomogeneity of topology, divide the transmission into intra-...

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...

Covering a graph or a lattice with nonoverlapping dimers is a problem that has received considerable interest in areas, such as discrete mathematics, statistical physics, chemistry, and materials science. Yet, the problem of percolation on dimer-covered lattices has received little attention. In particular, percolation on lattices that are fully covered by nonoverlapping dimers has not evidentl...

2014 We consider a set of ideal chains, each with N beads (N ~ 1) inscribed on a d-dimensional periodic lattice. Different chains are uncorrelated : thus any lattice site may belong to more than one chain ; two chains are said to be connected if they have at least one site in common. This defines a percolation problem (where the variable is the fraction c of occupied sites) physically related t...