## Abstract

We study competing first passage percolation on graphs generated by the configuration model. At time 0, vertex 1 and vertex 2 are infected with the type 1 and the type 2 infection, respectively, and an uninfected vertex then becomes type 1 (2) infected at rate λ_{1} (λ2 ) times the number of edges connecting it to a type 1 (2) infected neighbor. Our main result is that, if the degree distribution is a power-law with exponent τ ⋯ (2, 3), then as the number of vertices tends to infinity and with high probability, one of the infection types will occupy all but a finite number of vertices. Furthermore, which one of the infections wins is random and both infections have a positive probability of winning regardless of the values of λ_{1} and λ_{2} . The picture is similar with multiple starting points for the infections.

Original language | English |
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Pages (from-to) | 2419-2453 |

Number of pages | 35 |

Journal | The Annals of Applied Probability |

Volume | 26 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Aug 2016 |

## Keywords

- Coexistence
- Competing growth
- Configuration model
- Continuous-time branching process.
- First passage percolation
- Random graphs