Imagine I cover an arbitrarily large plane with randomly placed points at some density $\rho$ s.t. the number of points in any randomly sampled area $A$ (of arbitrary shape and size) is $\approx A\*\rho$. We form a graph $G$ by connecting any pair of points with an edge if they are within unit length distance of one-another. Here, $G$ should be a sort of random unit disc graph. 


Provided $\rho$, what is the probability that a randomly sampled vertex has degree $v = {0,1,2,...}$? If I select a set of points in a rectangular or circular area $A$ (which should introduce some bias), what is the expected probability distribution for the degree of vertices in the chosen set?


Also, I've been reading through the literature on percolation, but a heads up for a good paper relevant to the aforementioned graph $G$ in my question would be much appreciated!



The probability distribution for vertex degree in a unit disc graph generated from random points on a plane
The random placement of points you described is called [Poisson point process](http://en.wikipedia.org/wiki/Point_process#Poisson_point_process) (in the limit). The resulting graph is called [Random Geometric Graph](http://en.wikipedia.org/wiki/Random_geometric_graph). Searches will yield a good crop of papers about these graphs.


The degree distribution (again, in the limit) is Poisson with parameter equal to the expected number of points in a disc of radius 1.


