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Asymptotic normality of the $k$-core in random graphs

Svante Janson·Malwina J. Luczak
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Abstract

We study the $k$-core of a random (multi)graph on $n$ vertices with a given degree sequence. In our previous paper [Random Structures Algorithms 30 (2007) 50--62] we used properties of empirical distributions of independent random variables to give a simple proof of the fact that the size of the giant $k$-core obeys a law of large numbers as ${n\to \infty}$. Here we develop the method further and show that the fluctuations around the deterministic limit converge to a Gaussian law above and near the threshold, and to a non-normal law at the threshold. Further, we determine precisely the location of the phase transition window for the emergence of a giant $k$-core. Hence, we deduce corresponding results for the $k$-core in $G(n,p)$ and $G(n,m)$.

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Svante Janson, Malwina J. Luczak. 2008-06-27. Asymptotic normality of the $k$-core in random graphs. https://doi.org/10.1214/07-aap478

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