Studying the end-to-end distance of a self-avoiding or weakly self-avoiding random walk in two dimensions is a well known hard problem in probability and statistical physics. The conjecture is that the average end-to-end distance up to time \(n\) should be about \(n^{3/4}\).

It would seem that studying more complicated models would be even harder, but we are able to make progress in one such model. A star polymer is a collection of \(N\) weakly mutually-avoiding Brownian motions taking values in \(\mathbf{R}^d\) and starting at the origin. We study the two and three dimensional cases, and our sharpest results are for \(d=2\). Instead of the end-to-end distance, we define a radius \(R_T\) which measures the spread of the entire configuration up to time \(T\). There are two phases: a crowded phase for small values for \(T\), and a sparser phase for large \(T\) where paths do not interfere much. Our main result states for \(T<N\) (the crowded phase), that the radius \(R_T\) is approximately proportional to \(N^{1/4}T^{3/4}\).