Mathematics – Probability
Scientific paper
2010-12-10
Mathematics
Probability
The results are extended to random simple graphs in the finite third moment case
Scientific paper
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 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.
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