## Abstract

We introduce a formula for translating any upper bound on the percolation threshold of a lattice G into a lower bound on the exponential growth rate of lattice animals a(G) and vice versa. We exploit this in both directions. We obtain the rigorous lower bound pc(Z3 ) > 0.2522 for 3-dimensional site percolation. We also improve on the best known asymptotic bounds on a(Zd ) as d → ∞. Our formula remains valid if instead of lattice animals we enumerate certain subspecies called interfaces. Enumerating interfaces leads to functional duality formulas that are tightly connected to percolation and are not valid for lattice animals, as well as to strict inequalities for the percolation threshold. Incidentally, we prove that the rate of the exponential decay of the cluster size distribution of Bernoulli percolation is a continuous function of p ϵ (0, 1).

Original language | English |
---|---|

Pages (from-to) | 912-955 |

Number of pages | 44 |

Journal | Combinatorics Probability and Computing |

Volume | 32 |

Issue number | 6 |

Early online date | 31 Jul 2023 |

DOIs | |

Publication status | Published - 31 Jul 2023 |

### Bibliographical note

Publisher Copyright:© The Author(s), 2023. Published by Cambridge University Press.

## Keywords

- growth rate
- interface
- Lattice animals
- percolation

## ASJC Scopus subject areas

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics