Large-Scale Properties and Collapse Transition of Branched Polymers: Exact Results on Fractal Lattices

The asymptotic properties of branched polymers are studied on the two- and three-dimensional Sierpinski gaskets, with use of exact recursion equations. It is shown that loops are irrelevant on large scales and the exponents $\theta$ and $\nu$ for lattice animals are obtained exactly. In the presence of self-interactions, a collapse transition occurs at a nonzero critical temperature. At the transition the value ${\nu }_t$ of the gyration-radius exponent is very close to its value in the compact phase, in analogy with recent numerical results on two-dimensional branched polymers.