The bond percolation problem is studied by the Monte Carlo method on a two-dimensional square lattice of 2 X lo6 bonds. Through the inclusion of a ghost field h, we obtain the generating function (the percolation analogue of the Gibbs free energy), percolation probability (the analogue of the spontaneous magnetisation), and mean cluster size (‘isothermal susceptibility’) as functions of two ‘th...

The percolation problem is solved exactly in one dimension. The functions obtained bear a strong resemblance to those of the n-vector model on the same lattice. Further, we include a ghost field exactly in all dimensions d, thereby treating the ‘thermodynamics’ of percolation without appealing to the Potts model. In particular, we show for d = 1 that the nature of the singularities near the cri...

The onset of electrical percolation in nanotube-reinforced composites is often modeled by considering the geometric percolation of a system of penetrable, straight, rigid, capped cylinders, or spherocylinders, despite the fact that embedded nanotubes are not straight and do not penetrate one another. In Part I of this work we investigated the applicability of the soft-core model to the present ...

We consider the problem of distinguishing between different rates of percolation under noise. A statistical model of percolation is constructed allowing for the birth and death of edges as well as the presence of noise in the observations. This graph-valued stochastic process is composed of a latent and an observed nonstationary process, where the observed graph process is corrupted by type-I a...

In the strict Majority Bootstrap Percolation process each passive vertex v becomes active if at least ⌈ 2 ⌉ of its neighbors are active (and thereafter never changes its state). We address the problem of finding graphs for which a small proportion of initial active vertices is likely to eventually make all vertices active. We study the problem on a ring of n vertices augmented with a “central” ...

Scaling theory, duality symmetry, and numerical simulations of a random network model are used to study the magnetoresistance of a metal/insulator/perfect conductor composite with a disordered columnar microstructure. The phase diagram is found to have a critical line which separates regions of saturating and nonsaturating magnetoresistance. The percolation problem which describes this line is ...

This contribution is concerned with the effective viscosity problem, that is, homogenization of steady Stokes system a random array rigid particles, for which main difficulty treatment close particles. Standard approaches in literature have addressed this issue by making moment assumptions on interparticle distances. Such assumptions, however, prevent clustering not compatible physically-releva...