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

Abstract

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 $\ensuremath{\theta}$ and $\ensuremath{\nu}$ 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 ${\ensuremath{\nu}}_{t}$ 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.