Lattice animals: dc=8 for trees
Abstract
The asymptotic behavior of the total number of treelike clusters (lattice trees) on the hypercubic system is investigated for d>=4 by the method of series expansions. Interest centers on ascertaining the critical dimension, dc, at which the prefactor exponent θ attains its mean-field value. We present results for θ and for the cluster growth parameter λ for d=4 to 9. The λ values are close to within a few percent of those found for the general animal case. Results for θ have large uncertainties in the dimensions of interest, and the mean-field value is approached very gradually with d, so that the possibility of a lower value for dc cannot be completely discarded. Nevertheless, the available evidence suggests that dc=8 for lattice trees. This supports the findings of Lubensky and Isaacson in their recent work on lattice animals and dilute branched polymers.
- Publication:
-
Physical Review A
- Pub Date:
- September 1982
- DOI:
- 10.1103/PhysRevA.26.1791
- Bibcode:
- 1982PhRvA..26.1791R