The probability of unusually large components in the near-critical Erd\H{o}s-R\'enyi graph

Matthew I. Roberts

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arXiv · 1610.05485

The probability of unusually large components in the near-critical Erd\H{o}s-R\'enyi graph

Matthew I. Roberts

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Abstract

The largest components of the critical Erd\H{o}s-R\'enyi graph, $G(n,p)$ with $p=1/n$, have size of order $n^{2/3}$ with high probability. We give detailed asymptotics for the probability that there is an unusually large component, i.e. of size $an^{2/3}$ for large $a$. Our results, which extend work of Pittel, allow $a$ to depend upon $n$ and also hold for a range of values of $p$ around $1/n$. We also provide asymptotics for the distribution of the size of the component containing a particular vertex.

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Matthew I. Roberts. 2016-10-18. The probability of unusually large components in the near-critical Erd\H{o}s-R\'enyi graph. https://arxiv.org/abs/1610.05485

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