Open Access
2017 Critical window for the configuration model: finite third moment degrees
Souvik Dhara, Remco van der Hofstad, Johan S.H. van Leeuwaarden, Sanchayan Sen
Electron. J. Probab. 22: 1-33 (2017). DOI: 10.1214/17-EJP29


We investigate the component sizes of the critical configuration model, as well as the related problem of critical percolation on a supercritical configuration model. We show that, at criticality, the finite third moment assumption on the asymptotic degree distribution is enough to guarantee that the sizes of the largest connected components are of the order $n^{2/3}$ and the re-scaled component sizes (ordered in a decreasing manner) converge to the ordered excursion lengths of an inhomogeneous Brownian Motion with a parabolic drift. We use percolation to study the evolution of these component sizes while passing through the critical window and show that the vector of percolation cluster-sizes, considered as a process in the critical window, converge to the multiplicative coalescent process in the sense of finite dimensional distributions. This behavior was first observed for Erdős-Rényi random graphs by Aldous (1997) and our results provide support for the empirical evidences that the nature of the phase transition for a wide array of random-graph models are universal in nature. Further, we show that the re-scaled component sizes and surplus edges converge jointly under a strong topology, at each fixed location of the scaling window.


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Souvik Dhara. Remco van der Hofstad. Johan S.H. van Leeuwaarden. Sanchayan Sen. "Critical window for the configuration model: finite third moment degrees." Electron. J. Probab. 22 1 - 33, 2017.


Received: 8 May 2016; Accepted: 26 January 2017; Published: 2017
First available in Project Euclid: 15 February 2017

zbMATH: 06691463
MathSciNet: MR3622886
Digital Object Identifier: 10.1214/17-EJP29

Primary: 05C80 , 60C05

Keywords: Brownian excursions with parabolic drift , Critical configuration model , finite third moment degree , multiplicative coalescent , scaling window , Universality

Vol.22 • 2017
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