You might look at Robert MacPherson and Benjamin Schweinhart's recent preprint "Measuring Shape with Topology", where they use topological methods (i.e. persistent homology) to estimate fractal dimension for branched polymers, Brownian trees, and self-avoiding random walks.

Link: http://front.math.ucdavis.edu/1011.2258

You might look at Robert MacPherson and Benjamin Schweinhart's recent preprint "Measuring Shape with Topology", where they use topological methods (i.e. persistent homology) to estimate fractal dimension for branched polymers, Brownian trees, and self-avoiding random walks.

Link: https://arxiv.org/abs/1011.2258