Journal of Statistical Physics

, Volume 67, Issue 5, pp 1009–1038

The number and size of branched polymers in high dimensions

  • Takashi Hara
  • Gordon Slade

DOI: 10.1007/BF01049008

Cite this article as:
Hara, T. & Slade, G. J Stat Phys (1992) 67: 1009. doi:10.1007/BF01049008


We consider two models of branched polymers (lattice trees) on thed-dimensional hypercubic lattice: (i)the nearest-neighbor model in sufficiently high dimensions, and (ii) a “spread-out” or long-range model ford>8, in which trees are constructed from bonds of length less than or equal to a large parameterL. We prove that for either model the critical exponent θ for the number of branched polymers exists and equals 5/2, and that the critical exponentv for the radius of gyration exists and equals 1/4. This improves our earlier results for the corresponding generating functions. The proof uses the lace expansion, together with an analysis involving fractional derivatives which has been applied previously to the self-avoiding walk in a similar context.

Key words

Branched polymers lattice trees lattice animals lace expansion critical exponents 

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • Takashi Hara
    • 1
  • Gordon Slade
    • 2
  1. 1.Department of Applied PhysicsTokyo Institute of TechnologyTokyoJapan
  2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

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