Journal of Statistical Physics

, Volume 67, Issue 5, pp 1009-1038

First online:

The number and size of branched polymers in high dimensions

  • Takashi HaraAffiliated withDepartment of Applied Physics, Tokyo Institute of Technology
  • , Gordon SladeAffiliated withDepartment of Mathematics and Statistics, McMaster University

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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