In this paper the efficient space virtualisation for Hoshen–Kopelman algorithm is presented. We observe minimal parallel overhead during computations, due to negligible communication costs. The proposed algorithm is applied for computation of random-site percolation thresholds for four dimensional simple cubic lattice with sites’ neighbourhoods containing next-next-nearest neighbours (3NN). The...

The r-neighbour bootstrap process on a graph G starts with an initial set A0 of “infected” vertices and, at each step of the process, a healthy vertex becomes infected if it has at least r infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of G eventually becomes infected, then we say that A0 percolates. We prove a conjecture of Balogh and Bollobá...

We study a new geometric bootstrap percolation model, line percolation, on the d-dimensional integer grid [n]. In line percolation with infection parameter r, infection spreads from a subset A ⊂ [n] of initially infected lattice points as follows: if there exists an axis-parallel line L with r or more infected lattice points on it, then every lattice point of [n] on L gets infected, and we repe...

The ’t Hooft criterion leading to confinement out of a percolating cluster of central vortices suggests defining a novel three-dimensional gauge theory directly on a random percolation process. Wilson loop is viewed as a counter of topological linking with the random clusters. Beyond the percolation threshold large Wilson loops decay with an area law and show the universal shape effects due to ...

Let r be a nonnegative real number. Attach a disc of radius r to infinitely many random points (including the origin). Lilypad percolation asks whether we can reach infinity from the origin by walking through ‘lilypads’, that is, moving from one disc to another only if the discs overlap. In this paper, we explain what we mean by infinitely many random points, giving a definition of a Poisson Ra...

We prove that two independent continuous-time simple random walks on the infinite open cluster of a Bernoulli bond percolation in the lattice Z meet each other infinitely many times. An application to the voter model is also discussed. 2000 MR subject classification: 60K

We study a competition model on Z where the two infections are driven by supercritical Bernoulli percolations with distinct parameters p and q. We prove that, for any q, there exist at most countably many values of p < min {q,− pc} such that coexistence can occur.

Formation of the macroscopically-infinite hydrogen-bonded water network in various aqueous systems occurs via 3D percolation transition when the probability of finding a spanning water cluster exceeds 95%. As a result, in a wide interval of water content below the percolation threshold, rarefied quasi-2D water networks span over the mesoscopic length scale. Formation and topology of spanning wa...

Atomic clusters have been deposited between lithographically defined contacts with nanometer scale separations. The design of the contacts is based on an appropriate application of percolation theory to conduction in cluster deposited devices and allows finite-size effects to be clearly observed. It is demonstrated, both by experiment and by simulation, that for small contact separations the pe...

Percolation theory concerns the emergence of connected clusters that percolate through a networked system. Previous studies ignored the effect that a node outside the percolating cluster may actively induce its inside neighbours to exit the percolating cluster. Here we study this inducing effect on the classical site percolation and K-core percolation, showing that the inducing effect always ca...