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 study combinatorial parameters of a recently introduced bootstrap percolation problem in finite projective planes. We present sharp results on the size of the minimum percolating sets and the maximal non-percolating sets. Additional results on the minimal and maximal percolation time as well as on the critical probability in the projective plane are also presented.

A problem of the crossover from percolation to diffusion transport is considered. A general scaling theory is proposed. It introduces phenomenologically four critical exponents which are connected by two equations. One exponent is completely new. It describes the increase of the diffusion below percola-tion threshold. As an example, an exact solution of one dimensional lattice problem is given....

This article reviews some effects of disorder in percolation systems away from the critical density pc. For densities below pc, the statistics of large clusters defines the animals problem. Its relation to the directed animals problem and the Lee–Yang edge singularity problem is described. Rare compact clusters give rise to Griffiths singularities in the free energy of diluted ferromagnets, and...

The Scaling Law for the Discrete Kinetic Growth Percolation Model The critical exponent of the total number of finite clusters α is calculated directly without using scaling hypothesis both below and above the percolation threshold pc based on a kinetic growth percolation model in two and three dimensions. Simultaneously, we can calculate other critical exponents β and γ, and show that the scal...

Percolation refers to the emergence of a giant connected cluster in a disordered system when the number of connections between nodes exceeds a critical value. The percolation phase transitions were believed to be continuous until recently when, in a new so-called "explosive percolation" problem for a competition-driven process, a discontinuous phase transition was reported. The analysis of evol...

We study the effects of adding loops to a critical percolation cluster on the diffusional, and equivalently, (scalar) elastic properties of the fractal network. From the numerical calculations of the eigenspectrum of the transition probability matrix, we find that the spectral dimension ds and the walk dimension dw change suddenly as soon as the floppy ends of a critical percolation cluster are...

We study the problem of wireless network resilience to node failures from a percolation-based perspective. In practical wireless networks, it is often the case that the failure probability of a node depends on its degree (number of neighbors). We model this phenomenon as a degree-dependent site percolation process on random geometric graphs. Due to its non-Poisson structure, degree-dependent si...

we introduce a model for prediction the behavior of electrodes which modified withcarbon nanotubes in a polymer medium. these kinds of polymer composites aredeveloped in recent years, and experimental data for its percolation threshold isavailable. we construct a model based on percolation theory and fractal dimensionsand using experimental percolation threshold for calculating the moments of c...

The probability distributions of the masses of the clusters spanning from top to bottom of a percolating lattice at the percolation threshold are obtained in all dimensions from two to five. The first two cumulants and the exponents for the universal scaling functions are shown to have simple power law variations with the dimensionality. The cases where multiple spanning clusters occur are disc...