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

TL;DR

This paper establishes that the growth constants for lattice trees and animals on high-dimensional integer lattices asymptotically approach 2d, with their critical functions converging to e, using elementary proofs based on prior lace expansion results.

Contribution

It provides a simplified proof of the asymptotic behavior of growth constants for lattice trees and animals in high dimensions, extending previous results.

Findings

Growth constants asymptotic to 2de as dimension increases

Critical one-point functions converge to e in high dimensions

Results hold for both nearest-neighbour and spread-out models

Abstract

We prove that the growth constants for nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice Zd 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.