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External DLA on a spanning-tree-weighted random planar map

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RGMW06 - RGM follow up

External diffusion limited aggregation (DLA) is a widely studied subject in the physics literature, with many manifestations in nature; but it is not well-understood mathematically in any environment. We consider external DLA on an infinite spanning-tree-weighted random planar map. We prove that the growth exponent for the external diameter of the DLA cluster exists and is equal to $2/d }$, where $d{\sqrt{2}}$ denotes the ``fractal dimension of $\sqrt{2}$-Liouville quantum gravity (LQG)''—-or, equivalently, the ball volume growth exponent for the spanning-tree weighted map. Our proof is based on the fact that the complement of an external DLA cluster on a spanning-tree weighted map is a spanning-tree weighted map with boundary, which allows us to reduce our problem to proving certain estimates for distances in random planar maps with boundary. This is joint work with Ewain Gwynne.




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