Phase transitions and percolation at criticality in enhanced random connection models

We study 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 $\mathcal{P}_{\lambda}$ in $\mathbb{R}^2$ of intensity $\lambda$. In the homogenous RCM, the vertices at $x,y$ are connected with probability $g(|x-y|)$, independent of everything else, where $g:[0,\infty) \to [0,1]$ and $| \cdot |$ is the Euclidean norm. In the inhomogenous version of the model, points of $\mathcal{P}_{\lambda}$ are endowed with weights that are non-negative independent random variables with distribution $P(W>w)= w^{-\beta}1_{[1,\infty)}(w)$, $\beta>0$. Vertices located at $x,y$ with weights $W_x,W_y$ are connected with probability $1 - \exp\left( - \frac{\eta W_xW_y}{|x-y|^{\alpha}} \right)$, $\eta, \alpha > 0$, independent of all else. The graphs are enhanced by considering the edges of the graph as straight line segments starting and ending at points of $\mathcal{P}_{\lambda}$. 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 orientation with the mid point of each segment located at a distinct point of $\mathcal{P}_{\lambda}$. Intersecting lines form a path in the graph. A graph is said to percolate if there is an infinite connected component or path. We derive conditions for the existence of a phase transition and show that there is no percolation at criticality.