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In 2002 Carleson and Makarov defined the \emph{geodesic Laplacian path model} (LPM), which describes the simultaneous growth of $k$ disjoint curves along geodesics in the complement of the curves. I will define the ALE particle aggregation model, whose scaling limit was shown by Sola, Turner and Viklund in 2019 to be a geodesic in the complement of the unit disc (i.e. a straight line), and extend this scaling limit result to the same setting as the Laplacian path model. We will see that started from $k$ large slits, the scaling limit of the ALE cluster is described exactly by the LPM, and I will also discuss the case when the initial conditions are perturbed by a small slit.

AGRS Research Seminar Series: Convergence Of A Particle Aggregation Cluster To The Geodesic Laplacian Path Model