Branched Polymers

Intended as models in chemistry or biology, branched polymers are often modeled, in turn, by lattice animals (trees on a grid); see, e.g., [3, 5, 8, 10, 18, 19]. However, continuum polymers turn out to be in some respects more tractable than their grid cousins. In order to study the behavior of branched polymers, and in particular to define and understand what random examples look like, we must define a parametrization and then attempt to compute, using that parametrization, volumes of various configuration spaces. In principle, we could then compute (say) the probability that a branched polymer of a particular size in a given dimension takes the form of a specific tree, or has diameter exceeding some number; and we could perhaps generate uniformly random examples in an efficient manner. Fortunately, the space of branched polymers of order n and dimension D posesses an obvious and natural parametrization. One of several equivalent ways to describe it is to specify the tree-type of the polymer, together with the n−1 D-dimensional angles at which