## Electronic Journal of Probability

### Critical window for the configuration model: finite third moment degrees

#### Abstract

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.

#### Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 16, 33 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1487127644

Digital Object Identifier
doi:10.1214/17-EJP29

Mathematical Reviews number (MathSciNet)
MR3622886

Zentralblatt MATH identifier
06691463

#### Citation

Dhara, Souvik; van der Hofstad, Remco; van Leeuwaarden, Johan S.H.; Sen, Sanchayan. Critical window for the configuration model: finite third moment degrees. Electron. J. Probab. 22 (2017), paper no. 16, 33 pp. doi:10.1214/17-EJP29. https://projecteuclid.org/euclid.ejp/1487127644

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