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!