External diffusion-limited aggregation on a spanning-tree-weighted random planar map

Abstract

Let M be the infinite spanning-tree-weighted random planar map, which is the local limit of finite random planar maps sampled with probability proportional to the number of spanning trees they admit. We show that a.s. the M-graph-distance diameter of the external diffusion-limited aggregation (DLA) cluster on M run for m steps is of order ${\mathit{m}}^{2/\mathit{d}+{\mathit{o}}_{\mathit{m}}(1)}$, where d is the metric ball volume growth exponent for M (which was shown to exist by Ding and Gwynne (Comm. Math. Phys. 374 (2020) 1877–1934). By known bounds for d, one has $0.55051\cdots \le 2/\mathit{d}\le 0.563315\dots$.

Along the way, we also prove that loop-erased random walk (LERW) on M typically travels graph distance ${\mathit{m}}^{2/\mathit{d}+{\mathit{o}}_{\mathit{m}}(1)}$ in m units of time and that the graph-distance diameter of a finite spanning-tree-weighted random planar map with n edges, with or without boundary, is of order ${\mathit{n}}^{1/\mathit{d}+{\mathit{o}}_{\mathit{n}}(1)}$ except on an event with probability decaying faster than any negative power of n.

Our proofs are based on a special relationship between DLA and LERW on spanning-tree-weighted random planar maps as well as estimates for distances in such maps which come from the theory of Liouville quantum gravity.