Random growth and SLE limits

We consider a particular growth process in the complex plane. Beginning with an initial seed, we attach a new particle to the nth cluster according to a probability density that is some negative power of harmonic measure - i.e. particles are more likely to attach themselves in areas unlikely to be hit by Brownian motion started from infinity.
If a conformal growth model with particles attached according to harmonic measure can be thought of as a colony growing to gather resources from its environment, we can think of this process as modelling a colony growing in a hostile environment, where each new member wants to minimise its exposure to the elements.

Some heuristic reasoning (supported by early calculations) suggests that in the extreme case (where the site of each new particle is chosen only between the points of minimum density), as the size of each particle decreases (keeping the total capacity the same), the process converges to an SLE4.

In this project, we hope to prove this result for the extreme case described, and extend it to the case of a continuous density for each new particle for a sufficiently large negative parameter.