On the exponential growth rates of lattice animals and interfaces, and new bounds on $p_c$

We point out 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 $\dot{p}_c(\mathbb{Z}^3) > 0.2522$ for 3-dimensional site percolation. We also 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 $p_c(\mathbb{Z}^d)$. Motivated by the above and other recent work in percolation theory, we study the exponential growth rate $b_r$ of the number of lattice `interfaces' of a given size as a function of their surface-to-volume ratio $r$. We prove that the values of the percolation parameter $p$ for which the interface size distribution has an exponential tail are uniquely determined by $b_r$ by comparison with a dimension-independent function $f(r):= \frac{(1+r)^{1+r}}{r^r}$. Incidentally, we prove that the rate of the exponential decay of the cluster size distribution of Bernoulli percolation is a continuous function of $p\in (0,1)$.