Combinatorics, Probability and Computing

Paper

Expansion in High Dimension for the Growth Constants of Lattice Trees and Lattice Animals

YURI MEJÍA MIRANDAa1 and GORDON SLADEa1

a1 Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2 (e-mail: amie.yuri@gmail.com, slade@math.ubc.ca)

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 , with rigorous error estimates. The proof uses the lace expansion, together with a new expansion for the one-point functions based on inclusion–exclusion.

(Received August 16 2012)

(Revised March 01 2013)

(Online publication April 15 2013)

2010 Mathematics subject classification:

  • Primary 60K35;
  • Secondary 82B41

Footnotes

  Supported in part by CONACYT of Mexico.

  Supported in part by NSERC of Canada.