The localization transition and the critical properties of the Lorentz model in three dimensions are investigated by computer simulations. We give a coherent and quantitative explanation of the dynamics in terms of continuum percolation theory and obtain an excellent matching of the critical density and exponents. Within a dynamic scaling Ansatz incorporating two divergent length scales we achi...

Three-dimensional bond or site percolation theory on a lattice can be interpreted as a gauge theory in which the Wilson loops are viewed as counters of topological linking with random clusters. Beyond the percolation threshold large Wilson loops decay with an area law and show the universal shape effects due to flux tube quantum fluctuations like in ordinary confining gauge theories. Wilson loo...

Many different concepts of percolation exist for networks. We show that bond percolation, site percolation, k-core percolation, and bootstrap percolation are all special cases of the Watts Threshold model. We show that the “heterogeneous k-core” and a corresponding heterogeneous bootstrap model are equivalent to one another and the Watts Threshold Model. A more recent model of a “Generalized Ep...

A new algorithm for the derivation of low-density series for percolation on directed lattices is introduced and applied to the square lattice bond and site problems. Numerical evidence shows that the computational complexity grows exponentially, but with a growth factor λ < 8 √ 2, which is much smaller than the growth factor λ = 4 √2 of the previous best algorithm. For bond (site) percolation o...

We analyze a mathematical model of a cognitive radio network introduced in Yemeni et al. (2016). Our analysis reveals several surprising features of the model. We explain some of these features using ideas from percolation theory and stochastic geometry.

We report on a possible crossover of a non universal quantity at the upper critical dimensionality in the field of percolation. Plotting recent estimates for site percolation thresholds of hypercubes in dimension 6 ≤ d ≤ 13 against corresponding predictions from the GM formula pc = p0[(d − 1)(q − 1)] d for percolation thresholds, a significant departure of pc is observed for d ≥ 6. This result ...

It was a common belief until a very recent time that on two-dimensional (2D) lattices at percolation threshold pc there exists exactly one percolation cluster [1,2]. New insight developed recently by Aizenman, who proved [3] that the number of Incipient Spanning Clusters (ISC) in 2D critical percolation can be larger than one, and that the probability of at least k separate clusters is bounded ...

We explicitly construct an uncountable class of infinite aperiodic plane graphs which have equal, and explicitly computable, bond percolation thresholds. Furthermore for both bond percolation and the random-cluster model all large scale properties, such as the values of the percolation threshold and the critical exponents, of the graphs are equal. This equivalence holds for all values of p and ...

Inhomogeneous percolation, for its closer relationship with real-life, can be more useful and reasonable than homogeneous percolation to illustrate the critical phenomena and dynamical behaviour of complex networks. However, due to its intricacy, the theoretical framework of inhomogeneous percolation is far from being complete and many challenging problems are still open. In this paper, we firs...

In this article, I give a pedagogical introduction and overview of percolation theory. Special emphasis will be put on the review of some of the most prominent of the algorithms that have been devised to study percolation numerically. At the central stage shall be the real-space renormalization group treatment of the percolation problem. As a rather novel application of this approach to percola...