# On the exponential growth rates of lattice animals and interfaces

## TL;DR

This paper establishes a new formula linking percolation thresholds and the exponential growth of lattice animals and interfaces, leading to improved bounds and insights into percolation properties.

## Contribution

It introduces a novel formula connecting percolation thresholds with lattice animal growth rates and extends these results to interfaces, enhancing understanding of percolation phenomena.

## Key findings

- Improved asymptotic bounds on lattice animal growth rates in high dimensions
- Derived functional duality formulas for interfaces related to percolation
- Proved continuity of the cluster size distribution decay rate in percolation

## 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 to improve on the best known asymptotic bounds on $a(\mathbb{Z}^d)$ as $d\to \infty$. Our formula remains valid if instead of lattice animals we enumerate certain sub-species 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 $p\in (0,1)$.