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July 2021 External diffusion-limited aggregation on a spanning-tree-weighted random planar map
Ewain Gwynne, Joshua Pfeffer
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Ann. Probab. 49(4): 1633-1676 (July 2021). DOI: 10.1214/20-AOP1486


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 m2/d+om(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 0.550512/d0.563315.

Along the way, we also prove that loop-erased random walk (LERW) on M typically travels graph distance m2/d+om(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 n1/d+on(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.

Funding Statement

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.


Download Citation

Ewain Gwynne. Joshua Pfeffer. "External diffusion-limited aggregation on a spanning-tree-weighted random planar map." Ann. Probab. 49 (4) 1633 - 1676, July 2021.


Received: 1 May 2019; Revised: 1 August 2020; Published: July 2021
First available in Project Euclid: 13 May 2021

Digital Object Identifier: 10.1214/20-AOP1486

Primary: 60J67, 82B24, 82B41

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 4 • July 2021
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