A probabilistic approach to the leader problem in random graphs

We study the fixation time of the identity of the leader, that is, the most massive component, in the general setting of Aldous's multiplicative coalescent, which in an asymptotic sense describes the evolution of the component sizes of a wide array of near-critical coalescent processes, including the classical Erdos-Renyi process. We show tightness of the fixation time in the "Brownian" regime, explicitly determining the median value of the fixation time to within an optimalO(1) window. This generalizes Luczak's result for the Erdos-Renyi random graph using completely different techniques. In the heavy-tailed case, in which the limit of the component sizes can be encoded using a thinned pure-jump Levy process, we prove that only one-sided tightness holds. This shows a genuine difference in the possible behavior in the two regimes.