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

Yuri Mejia Miranda (University of British Columbia)
Gordon Slade (University of British Columbia)

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.

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Pages: 129-136

Publication Date: February 25, 2011

DOI: 10.1214/ECP.v16-1612

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