Hostname: page-component-784d4fb959-mjqqg Total loading time: 0 Render date: 2025-07-15T02:23:31.664Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  15 April 2013

YURI MEJÍA MIRANDA  and
GORDON SLADE
YURI MEJÍA MIRANDA
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, CanadaV6T 1Z2 (e-mail: amie.yuri@gmail.com, slade@math.ubc.ca)
GORDON SLADE
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, CanadaV6T 1Z2 (e-mail: amie.yuri@gmail.com, slade@math.ubc.ca)
Get access Rights & Permissions [Opens in a new window]

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 $\mathbb{Z}^d$, with rigorous error estimates. The proof uses the lace expansion, together with a new expansion for the one-point functions based on inclusion–exclusion.

Keywords

Primary 60K35Secondary 82B41

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 4 , July 2013 , pp. 527 - 565
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Aleksandrowicz, G. and Barequet, G. (2012) The growth rate of high-dimensional tree polycubes. To appear in Electr. J. Combinatorics.Google Scholar
[2]Barequet, R., Barequet, G. and Rote, G. (2010) Formulae and growth rates of high-dimensional polycubes. Combinatorica 30 257–275.Google Scholar
[3]Borgs, C., Chayes, J. T., van der Hofstad, R. and Slade, G. (1999) Mean-field lattice trees. Ann. Combin. 3 205–221.Google Scholar
[4]Clisby, N., Liang, R. and Slade, G. (2007) Self-avoiding walk enumeration via the lace expansion. J. Phys. A: Math. Theor. 40 10973–11017.Google Scholar
[5]Derbez, E. and Slade, G. (1997) Lattice trees and super-Brownian motion. Canad. Math. Bull. 40 19–38.CrossRefGoogle Scholar
[6]Derbez, E. and Slade, G. (1998) The scaling limit of lattice trees in high dimensions. Comm. Math. Phys. 193 69–104.Google Scholar
[7]Fisher, M. E. and Gaunt, D. S. (1964) Ising model and self-avoiding walks on hypercubical lattices and ‘high-density’ expansions. Phys. Rev. 133 A224–239.CrossRefGoogle Scholar
[8]Gaunt, D. S. and Peard, P. J. (2000) 1/d-expansions for the free energy of weakly embedded site animal models of branched polymers. J. Phys. A: Math. Gen. 33 7515–7539.Google Scholar
[9]Gaunt, D. S., Peard, P. J., Soteros, C. E. and Whittington, S. G. (1994) Relationships between growth constants for animals and trees. J. Phys. A: Math. Gen. 27 7343–7351.Google Scholar
[10]Gaunt, D. S. and Ruskin, H. (1978) Bond percolation processes in d dimensions. J. Phys. A: Math. Gen. 11 1369–1380.Google Scholar
[11]Graham, B. T. (2010) Borel-type bounds for the self-avoiding walk connective constant. J. Phys. A: Math. Theor. 43 235001.CrossRefGoogle Scholar
[12]Hara, T. (2008) Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals. Ann. Probab. 36 530–593.Google Scholar
[13]Hara, T. and Slade, G. (1990) On the upper critical dimension of lattice trees and lattice animals. J. Statist. Phys. 59 1469–1510.Google Scholar
[14]Hara, T. and Slade, G. (1992) The number and size of branched polymers in high dimensions. J. Statist. Phys. 67 1009–1038.Google Scholar
[15]Hara, T. and Slade, G. (1995) The self-avoiding-walk and percolation critical points in high dimensions. Combin. Probab. Comput. 4 197–215.CrossRefGoogle Scholar
[16]Harris, A. B. (1982) Renormalized (1/σ) expansion for lattice animals and localization. Phys. Rev. B 26 337–366.Google Scholar
[17]van der Hofstad, R. and Sakai, A. (2005) Critical points for spread-out self-avoiding walk, percolation and the contact process. Probab. Theory Rel. Fields 132 438–470.Google Scholar
[18]van der Hofstad, R. and Slade, G. (2005) Asymptotic expansions in n −1 for percolation critical values on the n-cube and Zn. Random Struct. Alg. 27 331–357.Google Scholar
[19]van der Hofstad, R. and Slade, G. (2006) Expansion in n −1 for percolation critical values on the n-cube and Zn: The first three terms. Combin. Probab. Comput. 15 695–713.Google Scholar
[20]Holmes, M. (2008) Convergence of lattice trees to super-Brownian motion above the critical dimension. Electron. J. Probab. 13 671–755.Google Scholar
[21]Janse van Rensburg, E. J. (2000) The Statistical Mechanics of Interacting Walks, Polygons, Animals and Vesicles. Oxford University Press.Google Scholar
[22]Klarner, D. A. (1967) Cell growth problems. Canad. J. Math. 19 851–863.Google Scholar
[23]Klein, D. J. (1981) Rigorous results for branched polymer models with excluded volume. J. Chem. Phys. 75 5186–5189.Google Scholar
[24]Madras, N. (1999) A pattern theorem for lattice clusters. Ann. Combin. 3 357–384.Google Scholar
[25]Mejía Miranda, Y. (2012) The critical points of lattice trees and lattice animals in high dimensions. PhD thesis, University of British Columbia.Google Scholar
[26]Mejía Miranda, Y. and Slade, G. (2011) The growth constants of lattice trees and lattice animals in high dimensions. Electron. Comm. Probab. 16 129–136.Google Scholar
[27]Peard, P. J. and Gaunt, D. S. (1995) 1/d-expansions for the free energy of lattice animal models of a self-interacting branched polymer. J. Phys. A: Math. Gen. 28 6109–6124.Google Scholar
[28]Penrose, M. D. (1992) On the spread-out limit for bond and continuum percolation. Ann. Appl. Probab. 3 253–276.Google Scholar
[29]Penrose, M. D. (1994) Self-avoiding walks and trees in spread-out lattices. J. Statist. Phys.Google Scholar
[30]Slade, G. (2006) The Lace Expansion and its Applications, Vol. 1879 of Lecture Notes in Mathematics, Ecole d'Eté de Probabilités de Saint–Flour XXXIV–2004, Springer.Google Scholar