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The probability of unusually large components in the near-critical Erdős–Rényi graph

  • Matthew I. Roberts (a1)

Abstract

The largest components of the critical Erdős–Rényi 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 the work of Pittel (2001), 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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* Postal address: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. Email address: mattiroberts@gmail.com

References

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