A probabilistic approach to the leader problem in random graphs. Issue 1 (15th October 2020)

• Record Type: Journal Article
• Authors: Addario‐Berry, Louigi; Bhamidi, Shankar; Sen, Sanchayan
• Is Part Of: Random structures & algorithms. Volume 58:Issue 1(2021)
• Page Start: 34
• Page End: 67
• Publication Date: 2020-10-15
• Subjects: critical random graphs -- entrance boundary -- Erdős‐Rényi random graph -- inhomogeneous random graphs -- Markov processes -- multiplicative coalescent; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20966
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract : 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.