A model of the spatial emergence of an interstellar civilization into a uniform distribution of habitable systems is presented. The process of emigration is modelled as a three-dimensional probabilistic cellular automaton. An algorithm is presented which defines both the daughter colonies of the original seed vertex and all subsequent connected vertices, and the probability of a connection between any two vertices. The automaton is analysed over a wide set of parameters for iterations that represent up to 250 000 years within the model's assumptions. Emigration patterns are characterized and used to evaluate two hypotheses that aim to explain the Fermi Paradox. The first hypothesis states that interstellar emigration takes too long for any civilization to have yet come within a detectable distance, and the second states that large volumes of habitable space may be left uninhabited by an interstellar civilization and Earth is located in one of these voids.

Consider a homogeneous Poisson process in with density ρ, and add the origin as an extra point. Now connect any two points x and y of the process with probability g(x − y), independently of the point process and all other pairs, where g is a function which depends only on the Euclidean distance between x and y, and which is nonincreasing in the distance. We distinguish two critical densities in this model. The first is the infimum of all densities for which the cluster of the origin is infinite with positive probability, and the second is the infimum of all densities for which the expected size of the cluster of the origin is infinite. It is known that if , then the two critical densities are non-trivial, i.e. bounded away from 0 and ∞. It is also known that if g is of the form , for some r > 0, then the two critical densities coincide. In this paper we generalize this result and show that under the integrability condition mentioned above the two critical densities are always equal.

In a planar percolation model, faces of the underlying graph, as well as the sites and bonds, may be viewed as random elements. With this viewpoint, Whitney duality allows construction of a planar dual percolation model for each planar percolation model, which applies to mixed models with sites, bonds, and faces open or closed at random. Using self-duality for percolation models on the square lattice, information is obtained about the percolative region in the mixed model.

The two common critical probabilities for a lattice graph L are the cluster size critical probability pH(L) and the mean cluster size critical probability pT(L). The values for the honeycomb lattice H and the triangular lattice T are proved to be pH(H) = pT(H) = 1–2 sin (π/18) and PH(T) = pT(T) = 2 sin (π/18). The proof uses the duality relationship and the star-triangle relationship between the two lattices, to find lower bounds for sponge-crossing probabilities.