Branched polymers on fractal lattices

The asymptotic properties of large branched polymers placed on fractal lattices are studied using exact recursion relations. Simple examples are given on quasilinear fractals to illustrate the general method and explain how the critical exponents theta and ν are obtained. For more complex systems we find that a collapse transition may occur in the presence of attractive interactions between the indi- vidual monomers (i.e., for a polymer in a bad solvent). Above the critical temperature the polymer has the geometry of a random lattice animal, while below $T_c$ it shrinks into a compact globule state. The geometrical and thermal exponents at the transition are also obtained exactly and compared to recent work on regular Euclidean lattices. For one of the lattices studied we find an essential singularity in the polymer generating function, rather than a power-law singularity. This suggests that a similar behavior may also occur in other systems lacking translational invariance, such as polymers on randomly dilute lattices.