Tip growth in a strongly concentrated conformal growth model follows local geodesics

We analyse the aggregate Loewner evolution (ALE), introduced in 2018 by Sola, Turner and Viklund to generalise versions of diffusion limited aggregation (DLA) in the plane using complex analysis. They showed convergence of the ALE for certain parameters to a single growing slit. Started from a non-trivial initial configuration of k needles and the same parameters, we show that the small-particle scaling limit of ALE is the Laplacian path model, introduced by Carleson and Makarov, in which the tips grow along geodesics towards $\infty$ . Our proof involves analysis of Loewner's equation near its singular points, and we extend martingale methods to the backward equation, where what we have to control is non-adapted. Most conformal growth models introduce an extra regularisation factor to deal with the singularities in Loewner's equation at the sharp tips and right-angle bases of slit particles. As an intermediate step we prove a limit result for a model with no such regularisation factor, developing methods which should prove useful in analysing other weakly-regularised models with non-trivial limits.