Abstract. We study the phase transition in a random graph in which vertices and edges are added at constant rates. Two recent papers in Physical Review E by Callaway, Hopcroft, Kleinberg, Newman, and Strogatz, and Dorogostev, Mendes, and Samukhin have computed the critical value of this model, shown that the fraction of vertices in finite clusters is infinitely differentiable at the critical value, and that in the subcritical phase the cluster size distribution has a polynomial decay rate with a continuously varying power law. Here we sketch rigorous proofs for the first and third results and a new estimate about connectivity probabilities at the critical value.
Preprint of the paper to appear in the Proceedings of the conference Discrete Random Walks 2003 held Sept. 1-5 at the Institute Henri Poincare in Paris.