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 , 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 .
Along the way, we also prove that loop-erased random walk (LERW) on M typically travels graph distance 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 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.
The second author was partially supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1745302.
We thank two anonymous referees for helpful comments on an earlier version of this paper.
"External diffusion-limited aggregation on a spanning-tree-weighted random planar map." Ann. Probab. 49 (4) 1633 - 1676, July 2021. https://doi.org/10.1214/20-AOP1486