### Abstract

Recently, the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank-1 type having finite variance but infinite third moment degrees was obtained in Bhamidi et al. (Ann Probab 40:2299–2361, 2012). It was proved that when the degrees obey a power law with exponent τ∈ (3 , 4) , the sequence of clusters ordered in decreasing size and multiplied through by n^{-} ^{(} ^{τ} ^{-} ^{2} ^{)} ^{/} ^{(} ^{τ} ^{-} ^{1} ^{)} converges as n→ ∞ to a sequence of decreasing non-degenerate random variables. Here, we study the tails of the limit of the rescaled largest cluster, i.e., the probability that the scaling limit of the largest cluster takes a large value u, as a function of u. This extends a related result of Pittel (J Combin Theory Ser B 82(2):237–269, 2001) for the Erdős–Rényi random graph to the setting of rank-1 inhomogeneous random graphs with infinite third moment degrees. We make use of delicate large deviations and weak convergence arguments.

Original language | English |
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Pages (from-to) | 38-95 |

Number of pages | 58 |

Journal | Journal of Statistical Physics |

Volume | 171 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Apr 2018 |

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### Keywords

- Critical random graphs
- Exponential tilting
- Inhomogeneous networks
- Large deviations
- Power-law degrees
- Thinned Lévy processes