The growth constants of lattice trees and lattice animals in high dimensions

Abstract

We prove that the growth constants for nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice $\mathbb{Z}^d$ are asymptotic to $2de$ as the dimension goes to infinity, and that their critical one-point functions converge to $e$. Similar results are obtained in dimensions $d > 8$ in the limit of increasingly spread-out models; in this case the result for the growth constant is a special case of previous results of M. Penrose. The proof is elementary, once we apply previous results of T. Hara and G. Slade obtained using the lace expansion.