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December 2014 The component sizes of a critical random graph with given degree sequence
Adrien Joseph
Ann. Appl. Probab. 24(6): 2560-2594 (December 2014). DOI: 10.1214/13-AAP985

Abstract

Consider a critical random multigraph $\mathcal{G}_{n}$ with $n$ vertices constructed by the configuration model such that its vertex degrees are independent random variables with the same distribution $\nu$ (criticality means that the second moment of $\nu$ is finite and equals twice its first moment). We specify the scaling limits of the ordered sequence of component sizes of $\mathcal{G}_{n}$ as $n$ tends to infinity in different cases. When $\nu$ has finite third moment, the components sizes rescaled by $n^{-2/3}$ converge to the excursion lengths of a Brownian motion with parabolic drift above past minima, whereas when $\nu$ is a power law distribution with exponent $\gamma\in(3,4)$, the components sizes rescaled by $n^{-(\gamma-2)/(\gamma-1)}$ converge to the excursion lengths of a certain nontrivial drifted process with independent increments above past minima. We deduce the asymptotic behavior of the component sizes of a critical random simple graph when $\nu$ has finite third moment.

Citation

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Adrien Joseph. "The component sizes of a critical random graph with given degree sequence." Ann. Appl. Probab. 24 (6) 2560 - 2594, December 2014. https://doi.org/10.1214/13-AAP985

Information

Published: December 2014
First available in Project Euclid: 26 August 2014

zbMATH: 1318.60015
MathSciNet: MR3262511
Digital Object Identifier: 10.1214/13-AAP985

Subjects:
Primary: 05C80, 60C05, 90B15

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.24 • No. 6 • December 2014
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