### Abstract

We consider a model of long-range first-passage percolation on the d-dimensional square lattice Z^{d} in which any two distinct vertices x,y∈Z^{d} are connected by an edge having exponentially distributed passage time with mean x-y ^{α+o(1)}, where α>0 is a fixed parameter and . is the l^{1}-norm on Z^{d}. We analyze the asymptotic growth rate of the set ß_{t}, which consists of all x∈Z^{d} such that the first-passage time between the origin 0 and x is at most t as t→∞. We show that depending on the values of α there are four growth regimes: (i) instantaneous growth for α<^{d}, (ii) stretched exponential growth for α∈(d,2d), (iii) superlinear growth for α∈(2d,2d+1), and finally (iv) linear growth for α>2d+1 like the nearest-neighbor first-passage percolation model corresponding to α=∞.

Original language | English (US) |
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Pages (from-to) | 203-256 |

Number of pages | 54 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 69 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2016 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics