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 Erdős‐Rényi 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 Łuczak's result for the Erdős‐Rényi 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 Lévy process, we prove that only one‐sided tightness holds. This shows a genuine difference in the possible behavior in the two regimes.

中文翻译：

---

随机图中前导问题的概率方法

我们研究领导者身份的固定时间，即Aldous乘性合并的一般设置中最重要的部分的时间，从渐近的角度描述了一系列近临界合并的组件大小的演变流程，包括经典的Erdős-Rényi流程。我们展示了“布朗”体制中固定时间的紧密性，明确确定了固定时间的中值在最佳O之内（1）窗口。这使用完全不同的技术将Łuczak的结果推广到Erdős-Rényi随机图。在重尾情况下，可以使用变薄的纯跳跃Lévy过程对组件大小的限制进行编码，我们证明只有单侧紧度成立。这表明两种制度下可能行为的真正差异。