Polymers and random graphs

We establish a precise connection between gelation of polymers in Lushnikov's model and the emergence of the giant component in random graph theory. This is achieved by defining a modified version of the Erdös-Rényi process; when contracting to a polymer state space, this process becomes a discrete-time Markov chain embedded in Lushnikov's process. The asymptotic distribution of the number of transitions in Lushnikov's model is studied. A criterion for a general Markov chain to retain the Markov property under the grouping of states is derived. We obtain a noncombinatorial proof of a theorem of Erdös-Rényi type.