Partition functions of discrete coalescents: from Cayley's formula to Frieze's \zeta(3) limit theorem

In these expository notes, we describe some features of the multiplicative coalescent and its connection with random graphs and minimum spanning trees. We use Pitman’s proof of Cayley’s formula, which proceeds via a calculation of the partition function of the additive coalescent, as motivation and as a launchpad. We define a random variable which may reasonably be called the empirical partition function of the multiplicative coalescent, and show that its typical value is exponentially smaller than its expected value. Our arguments lead us to an analysis of the susceptibility of the Erd\H{o}s-R'enyi random graph process, and thence to a novel proof of Frieze’s \zeta(3)-limit theorem for the weight of a random minimum spanning tree.