On the exponential growth rates of lattice animals and interfaces II: new asymptotic bounds

Abstract

We introduce a method for translating any upper bound on the percolation threshold of a lattice $G$ into a lower bound on the exponential growth rate $a(G)$ of lattice animals and vice-versa. We exploit this in both directions. We improve on the best known asymptotic lower and upper bounds on $a({\mathbb{Z}}^d)$ as $d\to \infty$. We use percolation as a tool to obtain the latter, and conversely we use the former to obtain lower bounds on $p_c({\mathbb{Z}}^d)$. We obtain the rigorous lower bound ${\dot{p}}_c({\mathbb{Z}}^3)>0.2522$ for 3-dimensional site percolation.