Skip to main content

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


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.

Hide All
[1]Aleksandrowicz G. and Barequet G. (2012) The growth rate of high-dimensional tree polycubes. To appear in Electr. J. Combinatorics.
[2]Barequet R., Barequet G. and Rote G. (2010) Formulae and growth rates of high-dimensional polycubes. Combinatorica 30 257275.
[3]Borgs C., Chayes J. T., van der Hofstad R. and Slade G. (1999) Mean-field lattice trees. Ann. Combin. 3 205221.
[4]Clisby N., Liang R. and Slade G. (2007) Self-avoiding walk enumeration via the lace expansion. J. Phys. A: Math. Theor. 40 1097311017.
[5]Derbez E. and Slade G. (1997) Lattice trees and super-Brownian motion. Canad. Math. Bull. 40 1938.
[6]Derbez E. and Slade G. (1998) The scaling limit of lattice trees in high dimensions. Comm. Math. Phys. 193 69104.
[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 A224239.
[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 75157539.
[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 73437351.
[10]Gaunt D. S. and Ruskin H. (1978) Bond percolation processes in d dimensions. J. Phys. A: Math. Gen. 11 13691380.
[11]Graham B. T. (2010) Borel-type bounds for the self-avoiding walk connective constant. J. Phys. A: Math. Theor. 43 235001.
[12]Hara T. (2008) Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals. Ann. Probab. 36 530593.
[13]Hara T. and Slade G. (1990) On the upper critical dimension of lattice trees and lattice animals. J. Statist. Phys. 59 14691510.
[14]Hara T. and Slade G. (1992) The number and size of branched polymers in high dimensions. J. Statist. Phys. 67 10091038.
[15]Hara T. and Slade G. (1995) The self-avoiding-walk and percolation critical points in high dimensions. Combin. Probab. Comput. 4 197215.
[16]Harris A. B. (1982) Renormalized (1/σ) expansion for lattice animals and localization. Phys. Rev. B 26 337366.
[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 438470.
[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 331357.
[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 695713.
[20]Holmes M. (2008) Convergence of lattice trees to super-Brownian motion above the critical dimension. Electron. J. Probab. 13 671755.
[21]Janse van Rensburg E. J. (2000) The Statistical Mechanics of Interacting Walks, Polygons, Animals and Vesicles. Oxford University Press.
[22]Klarner D. A. (1967) Cell growth problems. Canad. J. Math. 19 851863.
[23]Klein D. J. (1981) Rigorous results for branched polymer models with excluded volume. J. Chem. Phys. 75 51865189.
[24]Madras N. (1999) A pattern theorem for lattice clusters. Ann. Combin. 3 357384.
[25]Mejía Miranda Y. (2012) The critical points of lattice trees and lattice animals in high dimensions. PhD thesis, University of British Columbia.
[26]Mejía Miranda Y. and Slade G. (2011) The growth constants of lattice trees and lattice animals in high dimensions. Electron. Comm. Probab. 16 129136.
[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 61096124.
[28]Penrose M. D. (1992) On the spread-out limit for bond and continuum percolation. Ann. Appl. Probab. 3 253276.
[29]Penrose M. D. (1994) Self-avoiding walks and trees in spread-out lattices. J. Statist. Phys.
[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.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 5 *
Loading metrics...

Abstract views

Total abstract views: 158 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 24th November 2017. This data will be updated every 24 hours.