The theoretical basis of continuum percolation has changed greatly since its beginning as little more than an analogy with lattice systems. Nevertheless, there is yet no comprehensive theory of this field. A basis for such a theory is provided here with the introduction of the Potts fluid, a system of interacting s-state spins which are free to move in the continuum. In the s → 1 limit, the Pot...

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...

In dynamical critical site percolation on the triangular lattice or bond percolation on Z , we define and study a local time measure on the exceptional times at which the origin is in an infinite cluster. We show that at a typical time with respect to this measure, the percolation configuration has the law of Kesten’s Incipient Infinite Cluster. In the most technical result of this paper, we sh...

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...

This paper deals with the critical point of three-dimensional bootstrap percolation-like cellular automata. Some general sufficient or necessary conditions for pc = 0 are obtained. In the case of pc > 0, some explicit upper and lower bounds are provided in terms of the critical value of oriented site percolation.

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...

The lattice fractal Sierpinski carpet and the percolation theory are applied to develop a new random stock price for the financial market. Percolation theory is usually used to describe the behavior of connected clusters in a random graph, and Sierpinski carpet is an infinitely ramified fractal. In this paper, we consider percolation on the Sierpinski carpet lattice, and the corresponding finan...

In 1982, Tsallis derived a formula which proposed an exact value of 0.522372078 . . . for the bond percolation threshold of the kagome lattice. We use the substitution method, which is based on stochastic ordering, to compare the probability distribution of connections in the homogeneous bond percolation model on the kagome lattice to those of an exactly-solved inhomogeneous bond percolation mo...