A probabilistic approach to the leader problem in random graphs

We study the fixation time of the identity of the leader, i.e., the most massive component, in the general setting of Aldous's multiplicative coalescent [4, 5], which in an asymptotic sense describes the evolution of the component sizes of a wide array of near-critical coalescent processes, including the classical Erd\H{o}s-R\'enyi process. We show tightness of the fixation time in the "Brownian" regime, explicitly determining the median value of the fixation time to within an optimal $O(1)$ window. This generalizes {\L}uczak's result [31] for the Erd\H{o}s-R\'enyi 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 L\'{e}vy process, we prove that only one-sided tightness holds. This shows a genuine difference in the possible behavior in the two regimes. The solution to the leader problem in the setting of the Erd\H{o}s-R\'enyi random graph played an important role in the study of the scaling limit of the minimal spanning tree on the complete graph [2]. We believe that analogous results, such as those proved herein, will be useful in establishing universality of the intrinsic geometry of the minimal spanning tree across a large class of models.