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2011 The growth constants of lattice trees and lattice animals in high dimensions
Yuri Mejia Miranda, Gordon Slade
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Electron. Commun. Probab. 16: 129-136 (2011). DOI: 10.1214/ECP.v16-1612


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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Yuri Mejia Miranda. Gordon Slade. "The growth constants of lattice trees and lattice animals in high dimensions." Electron. Commun. Probab. 16 129 - 136, 2011.


Accepted: 25 February 2011; Published: 2011
First available in Project Euclid: 7 June 2016

zbMATH: 1225.60154
MathSciNet: MR2775351
Digital Object Identifier: 10.1214/ECP.v16-1612

Primary: 60K35
Secondary: 82B41

Keywords: growth constant , lattice animal , lattice tree , mean-field model


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