Abstract
The number ofn-site lattice trees (up to translation) is believed to behave asymptotically asCn −0λn, where θ is a critical exponent dependent only on the dimensiond of the lattice. We present a rigorous proof that θ≥(d−1)/d for anyd≥2. The method also applies to lattice animals, site animals, and two-dimensional self-avoiding polygons. We also prove that θ≧v whend=2, wherev is the exponent for the radius of gyration.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Bovier, J. Fröhlich, and U. Glaus, Branched polymers and dimensional reduction, inCritical Phenomena, Random Systems, Gauge Theories, K. Osterwalder and R. Stora, eds. (North-Holland, Amsterdam, 1984).
D. S. Gaunt, M. F. Sykes, G. M. Torrie, and S. G. Whittington, Universality in branched polymers ond-dimensional hypercubic lattices,J. Phys. A: Math. Gen. 15:3209–3217 (1982).
T. Hara and G. Slade, On the upper critical dimension of lattice trees and lattice animals,J. Stat. Phys. 59:1469–1510 (1990).
T. Hara and G. Slade, Self-avoiding walks in five or more dimensions. I. The critical behaviour,Commun. Math. Phys. 147:101–136 (1992).
T. Hara and G. Slade, The number and size of branched polymers in high dimensions,J. Stat. Phys. 67:1009–1038 (1992).
J. Isaacson and T. C. Lubensky, Flory exponents for generalized polymer problems,J. Phys. Lett. 41:L469–471 (1980).
E. J. Janse van Rensburg, On the number of trees inZ d,J. Phys. A: Math. Gen. 25:3523–3528 (1992).
D. A. Klarner, Cell growth problems,Can. J. Math. 19:851–863 (1967).
D. J. Klein, Rigorous results for branched polymer models with excluded volume,J. Chem. Phys. 75:5186–5189 (1981).
L. H. Loomis and H. Whitney, An inequality related to the isoperimetric inequality,Bull. Am. Math. Soc. 55:961–962 (1949).
T. C. Lubensky and J. Isaacson, Statistics of lattice animals and dilute branched polymers,Phys. Rev. A 20:2130–2146 (1979).
N. Madras, Bounds on the critical exponent of self-avoiding polygons, inRandom Walks, Brownian Motion, and Interacting Particle Systems. R. Durrett and H. Kesten, eds. (Bikhäuser, Boston, 1991).
N. Madras and G. Slade,The Self-Avoiding Walk (Birkhäuser, Boston, 1993).
G. Parisi and N. Sourlas, Critical behavior of branched polymers and the Lee-Yang edge singularity,Phys. Rev. Lett. 46:871–874 (1981).
H. Tasaki and T. Hara, Critical behaviour in a system of branched polymers,Prog. Theor. Phys. Suppl. 92:14–25 (1987).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Madras, N. A rigorous bound on the critical exponent for the number of lattice trees, animals, and polygons. J Stat Phys 78, 681–699 (1995). https://doi.org/10.1007/BF02183684
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02183684