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 P λ in R 2 of intensity λ. In the homogeneous RCM, the vertices at x, y are connected with probability g(|x − y|), independent of everything else, where g : [0, ∞) → [0, 1] and |•| is the Euclidean norm. In the inhomogeneous version of the model, points of P λ are endowed with weights that are non-negative independent random variables with distribution P (W > w) = w −β 1 [1,∞) (w), β > 0. Vertices located at x, y with weights W x , W y are connected with probability 1 − exp − ηW x W y |x−y| α , η, α > 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 P λ . 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 P λ . 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.

We study phase transition and percolation at criticality for three planar random graph models, 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 P λ in R 2 of intensity λ. In the homogenous RCM, the vertices at x, y are connected with probability g(|x − y|), independent of everything else, where g : [0, ∞) → [0, 1] and | • | is the Euclidean norm. In the inhomogenous version of the model, points of P λ are endowed with weights that are non-negative independent random variables with distribution P (W > w) = w −β 1 [1,∞) (w), β > 0. Vertices located at x, y with weights Wx, Wy are connected with probability 1 − exp − ηWxWy |x−y| α , η, α > 0, independent of all else. The edges of the graph are viewed as straight line segments starting and ending at points of P λ . A path in the graph is a continuous curve that is a subset of the collection 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 line located at a distinct point of P λ . Intersecting lines then 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 under some additional conditions that there is no percolation at criticality.

We study an inhomogeneous random connection model in the connectivity regime. The vertex set of the graph is a homogeneous Poisson point process Ps of intensity s > 0 on the unit cubeEach vertex is endowed with an independent random weight distributed as W , whereGiven the vertex set and the weights an edge exists between x, y ∈ Ps with probability 1 − exp − ηWxWy (d(x,y)/r) α , independent of everything else, where η, α > 0, d(•, •) is the toroidal metric on S and r > 0 is a scaling parameter. We derive conditions on α, β such that under the scaling rs(ξ) d = 1 c 0 s log s + (k − 1) log log s + ξ + log αβ k!d , ξ ∈ R, the number of vertices of degree k converges in total variation distance to a Poisson random variable with mean e −ξ as s → ∞, where c0 is an explicitly specified constant that depends on α, β, d and η but not on k. In particular, for k = 0 we obtain the regime in which the number of isolated nodes stabilizes, a precursor to establishing a threshold for connectivity. We also derive a sufficient condition for the graph to be connected with high probability for large s. The Poisson approximation result is derived using the Stein's method.