## Abstract

The behavior of the random graph G(n,p) around the critical probability p_{c}=1/n is well understood. When p = (1 + O(n^{1/3}))p_{c} the components are roughly of size n^{2/3} and converge, when scaled by n^{-2/3}, to excursion lengths of a Brownian motion with parabolic drift. In particular, in this regime, they are not concentrated. When p = (1-ε(n))p_{c} with ε(n)n^{1/3} → ∞ (the subcritical regime) the largest component is concentrated around 2ε^{-2} log(ε^{3}n). When p = (1 + ε(n))p_{c} with ε(n)n^{1/3} → ∞ (the supercritical regime), the largest component is concentrated around 2εn and a duality principle holds: other component sizes are distributed as in the subcritical regime. Itai Benjamini asked whether the same phenomenon occurs in a random d-regular graph. Some results in this direction were obtained by (Pittel, Ann probab 36 (2008) 1359-1389). In this work, we give a complete affirmative answer, showing that the same limiting behavior (with suitable d dependent factors in the non-critical regimes) extends to random d-regular graphs.

Original language | English |
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Pages (from-to) | 111-148 |

Number of pages | 38 |

Journal | Random Structures and Algorithms |

Volume | 36 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2010 |

Externally published | Yes |

## Keywords

- Percolation
- Random regular graph