Expansion in high dimension for the growth constants of lattice trees and lattice animals

TL;DR

This paper derives the first three terms of the 1/d expansion for growth constants and one-point functions of lattice trees and animals in high dimensions, using lace expansion and inclusion-exclusion techniques.

Contribution

It introduces a new expansion method for one-point functions and provides rigorous error estimates for the high-dimensional asymptotics.

Findings

First three terms of 1/d expansion computed

Rigorous error bounds established

New expansion technique for one-point functions

Abstract

We compute the first three terms of the 1/d expansions for the growth constants and one-point functions of nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice Zd, with rigorous error estimates. The proof uses the lace expansion, together with a new expansion for the one-point functions based on inclusion-exclusion.