SLE scaling limits for a Laplacian random growth model

Abstract

We consider a model of planar random aggregation from the ALE$(0,η)$ family where particles are attached preferentially in areas of low harmonic measure. We find that the model undergoes a phase transition in negative $η$, where for sufficiently large values the attachment distribution of each particle becomes atomic in the small particle limit, with each particle attaching to one of the two points at the base of the previous particle. This complements the result of Sola, Turner and Viklund for large positive $η$, where the attachment distribution condenses to a single atom at the tip of the previous particle. As a result of this condensation of the attachment distributions we deduce that in the limit as the particle size tends to zero the ALE cluster converges to a Schramm--Loewner evolution with parameter $κ= 4$ (SLE$_4$). We also conjecture that using other particle shapes from a certain family, we have a similar SLE scaling result, and can obtain SLE$_κ$ for any $κ\geq 4$.