The no-enclave percolation (NEP) model introduced recently by Sheinman et al. can be mapped to a problem of holes within a standard percolation backbone, and numerical measurements of such holes give the same size-distribution exponent τ=1.82(1) as found for the NEP model. An argument is given that τ=1+d_{B}/2≈1.822 for backbone holes, where d_{B} is the backbone dimension. On the other hand, a...

We look at the process of inheritance of mitochondrial DNA as a percolation model on trees equivalent to the Galton-Watson process. The model is exactly solvable for its percolation threshold pc and percolation probability critical exponent. In the approximation of small percolation probability, and assuming limited progeny number, we are also able to find the maximum and minimum percolation pr...

Temperature data from the unsaturated zone (UZ) at Yucca Mountain are analyzed to estimate percolation-flux rates and overall heat flux. A multilayer, one-dimensional analytical solution is presented for determining percolation flux from temperature data. Case studies have shown that the analytical solution agrees very well with results from the numerical code, TOUGH2. The results of the analys...

We report an extensive finite-size study of polymer networks near the percolation threshold, using numerical techniques. The polymers are modeled by random walks occupying the bonds of a two-dimensional square lattice. We measure the percolation threshold and critical exponents of the networks for various polymer lengths. We find that the critical occupation probability is a decreasing function...

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 suitable interpolation between classical percolation and a special variant of explosive percolation enables the explicit realization of a tricritical percolation point. With high-precision simulations of the order parameter and the second moment of the cluster size distribution a fully consistent tricritical scaling scenario emerges yielding the tricritical crossover exponent 1/φ(t)=1.8 ± 0.1.

We reconsider the problem of percolation on an equilibrium random network with degree-degree correlations between nearest-neighboring vertices focusing on critical singularities at a percolation threshold. We obtain criteria for degree-degree correlations to be irrelevant for critical singularities. We present examples of networks in which assortative and disassortative mixing leads to unusual ...

We prove that AB site percolation occurs on the line graph of the square lattice when p ∈ (1− √ 1− pc, √ 1− pc), where pc is the critical probability for site percolation in Z. Also, we prove that AB bond percolation does not occur on Z for p = 1 2 .

We describe the percolation model and some of the principal results and open problems in percolation theory. We also discuss briefly the spectacular recent progress by Lawler, Schramm, Smirnov and Werner towards understanding the phase transition of percolation (on the triangular lattice). 2000 Mathematics Subject Classification: 60K35, 82B43.