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

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 $dto infty$. 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.