## The Annals of Applied Probability

### The component sizes of a critical random graph with given degree sequence

#### 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.

#### Article information

Source
Ann. Appl. Probab. Volume 24, Number 6 (2014), 2560-2594.

Dates
First available in Project Euclid: 26 August 2014

https://projecteuclid.org/euclid.aoap/1409058040

Digital Object Identifier
doi:10.1214/13-AAP985

Mathematical Reviews number (MathSciNet)
MR3262511

Zentralblatt MATH identifier
1318.60015

#### Citation

Joseph, Adrien. The component sizes of a critical random graph with given degree sequence. Ann. Appl. Probab. 24 (2014), no. 6, 2560--2594. doi:10.1214/13-AAP985. https://projecteuclid.org/euclid.aoap/1409058040

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