### Abstract

We study limit laws for simple random walks on supercritical long-range percolation clusters on Z^{d}, d ≥ 1. For the long range percolation model, the probability that two vertices x,y are connected behaves asymptotically as ||x - y||^{-s} 2 . When s ∈ (d, d + 1), we prove that the scaling limit of simple random walk on the infinite component converges to an α-stable Lévy process with α = s - d establishing a conjecture of Berger and Biskup [Probab. Theory Related Fields 137 (2007) 83-120]. The convergence holds in both the quenched and annealed senses. In the case where d = 1 and s > 2 we show that the simple random walk converges to a Brownian motion. The proof combines heat kernel bounds from our companion paper [Crawford and Sly Probab. Theory Related Fields 154 (2012) 753-786], ergodic theory estimates and an involved coupling constructed through the exploration of a large number of walks on the cluster.

Original language | English (US) |
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Pages (from-to) | 445-502 |

Number of pages | 58 |

Journal | Annals of Probability |

Volume | 41 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1 2013 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- Long rang percolation
- Random walk in random environment
- Stable process.

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## Cite this

*Annals of Probability*,

*41*(2), 445-502. https://doi.org/10.1214/12-AOP774