  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.
