## Abstract

We study independent long-range percolation on Z^{d} where the vertices x and y are connected with probability 1 − e^{−}β⃦x−^{y}^{⃦−d−α} for α > 0. Provided the critical exponents δ and 2 − η defined by δ = lim_{n}_{→∞ log}(_{P}^{−}_{βc}^{log}_{(}|_{K}^{(n)}_{0|≥n))} and 2 − η = lim_{x}_{→∞}^{log}(^{Pβc (0↔x))} + d exist, where K_{0} is the cluster containing the origin, we show that log(⃦x⃦) _{δ}_{≥} d + (α ∧ 1) and 2 − η ≥ α ∧ 1. d − (α ∧ 1) The lower bound on δ is believed to be sharp for d = 1, α ∈ [ ^{1}_{3} , 1) and for d = 2, α ∈ [ ^{2}_{3} , 1], whereas the lower bound on 2 − η is sharp for d = 1, α ∈ (0, 1), and for α ∈ (0, 1] for d > 1, and is not believed to be sharp otherwise. Our main tool is a connection between the critical exponents and the isoperimetry of cubes inside Z^{d}.

Original language | English |
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Pages (from-to) | 721-730 |

Number of pages | 10 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 60 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2024 |

## Keywords

- Critical exponents
- Long-range percolation
- Phase transition