Nearest neighbor and hard sphere models in continuum percolation

Abstract

Consider a Poisson process X in Rd with density 1. We connect each point of X to its k nearest neighbors by undirected edges. The number k is the parameter in this model. We show that, for k = 1, no percolation occurs in any dimension, while, for k = 2, percolation occurs when the dimension is sufficiently large. We also show that if percolation occurs, then there is exactly one infinite cluster. Another percolation model is obtained by putting balls of radius zero around each point of X and let the radii grow linearly in time until they hit another ball. We show that this model exists and that there is no percolation in the limiting configuration. Finally we discuss some general properties of percolation models where balls placed at Poisson points are not allowed to overlap (but are allowed to be tangent).

Numerical modeling of nanostructured materials

“…The first step in the scheme of these methods is to randomly generate the desired number of particles in the domain (the size of each particle is initially 0). Then, the size of each particle is either defined as the half of the distance from its center to the center of its closest particle [27] or by progressively enlarging this particle until it comes into contact with another one in the assembly [28]. These methods are particularly fast because the particle positions are generated randomly.…”

Percolation in the secrecy graph: Bounds on the critical probability and impact of power constraints

“…While doing this, we survey various methods that have been used for other continuum percolation models. They are from [4], and [10], on percolation in the Gilbert disc model, and from [1] and [6], on percolation in the k-nearest neighbour model.…”

Percolation in the secrecy graph

“…Remove all the red points in D(0, 3r). In the "local coupling" described in [7], such a new configuration has positive probability (conditional on the original configuration). But, in the new configuration, all n components have joined up, so there is only one infinite component.…”

“…Proof: (Sketch) The proof of [7] goes through, except that we compare with a much simpler branching process, namely the one where the offspring size distribution is geometric with mean 1/λ, as above. There is no need to have two types of offspring.…”