Spatial models for spread of an epidemic may be mapped onto bond percolation. We point out that with disorder in the strength of contacts between individuals patchiness in the spread of the epidemic is very likely, and the criterion for epidemic outbreak depends strongly on the disorder because the critical region of the corresponding percolation model is broadened. In some networks the percola...

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 present an improved version of an earlier Larson-type model of amphiphiles. Detailed descriptions of the algorithms are given and some situations where care must be taken are pointed out We use the model to calculate the condensation transition temperature of amphiphiles in a ternary mixture. The concentration percolation threshold and percolation critical exponent of arnphiphile clusters at...

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...

We study the directed last-passage percolation model on the planar integer lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles serve as boundary conditions for stationary last-passage percolation, define solutions to ...

A useful result about leftmost and rightmost paths in two dimensional bond percolation is proved. This result was introduced without proof in [5] in the context of the contact process in continuous time. As discussed here, it also holds for several related models, including the discrete time contact process and two dimensional site percolation. Among the consequences are a natural monotonicity ...

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...

One of the most well-known classical results for site percolation on the square lattice is the equation pc + p ∗ c = 1. In words, this equation means that for all values 6= pc of the parameter p the following holds: Either a.s. there is an infinite open cluster or a.s. there is an infinite closed ‘star’ cluster. This result is closely related to the percolation transition being sharp: Below pc ...