Phase transition at critically, poisson convergence of isolated vertices and connectivity in random connection models

Abstract

This work has two parts. The first part contains the study of phase transition and percolation at criticality for three random graph models on the plane, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a homogeneous Poisson point process V?V_\lambdaV?? of intensity ?\lambda?. In the homogeneous RCM, the vertices at x,yx, yx,y are connected with probability g(?x?y?)g(|x - y|)g(?x?y?), independent of everything else, where g:[0,?)?[0,1]g : [0, \infty) \to [0,1]g:[0,?)?[0,1] and ???|\cdot|??? is the Euclidean norm. In the inhomogeneous version of the model, points of V?V_\lambdaV?? are endowed with weights that are non-negative independent random variables WWW, where P(W>w)=p>0\mathbb{P}(W > w) = p > 0P(W>w)=p>0. Vertices located at x,yx, yx,y with weights Wx,WyW_x, W_yWx?,Wy? are connected with probability 1?exp?(??WxWy?x?y???)1 - \exp(-\eta W_x W_y |x - y|^{-\alpha})1?exp(??Wx?Wy??x?y???) for some ?,?>0\eta, \alpha > 0?,?>0, independent of all else. The edges of the graph are viewed as straight line segments starting and ending at points of V?V_\lambdaV??. A path in the graph is a continuous curve that is a subset of the union of all these line segments. The Poisson stick model consists of line segments of independent random lengths and orientations with the midpoint of each line located at a distinct point of V?V_\lambdaV??. Intersecting lines then form a path in the graph. A graph is said to percolate if there is an infinite connected component or path. The conditions for the existence of a phase transition have been derived. Under some additional conditions, it has been shown that there is no percolation at criticality. In the second part, we consider an inhomogeneous random connection model on a ddd-dimensional unit cube S=[?12,12]dS = [-\frac{1}{2}, \frac{1}{2}]^dS=[?21?,21?]d with the toroidal metric. The vertex set is the homogeneous Poisson point process VsV_sVs? of intensity s>0s > 0s>0. The vertices are equipped with i.i.d. weights WWW and the connection function as above. Under a suitable choice of scaling, it can be shown that the number of isolated vertices converges to a Poisson random variable as s??s \to \inftys??. We also derive a sufficient condition for the graph to be connected with high probability.