Directed distances in spanning-tree-decorated planar maps: exact exponent, scaling limit and universality
Abstract
We define a natural orientation on a spanning-tree-decorated planar map whereby, roughly speaking, each directed edge in the map is oriented to match the direction of the contour exploration of the spanning tree. We study directed distances (lengths of shortest directed paths) with respect to this orientation. We construct the Busemann function which measures directed distances to ∞ along a natural interface in the uniform infinite spanning-tree-decorated map. We show that this Busemann function, re-scaled appropriately, converges in law to a 3/2-stable Lévy process. We also show that in a uniform spanning-tree-decorated map with n edges, directed distances are typically of order n1/3. Using a strong coupling argument, we deduce analogous statements for directed distances in other random planar maps in the √2-Liouville quantum gravity (LQG) universality class, including uniform meandric systems and mated-CRT maps for γ=√2. These results give the scaling dimension for a hypothetical directed version of the √2-LQG metric. Our proof strategy is inspired by work of Borga and Gwynne (2025) on directed distances in bipolar-oriented triangulations.
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BibTeX
@misc{borga2026directed,
title = {{Directed distances in spanning-tree-decorated planar maps: exact exponent, scaling limit and universality}},
author = {Jacopo Borga and Ewain Gwynne and Yuanzheng Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22514},
eprint = {2609.22514},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22514},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Directed distances in spanning-tree-decorated planar maps: exact exponent, scaling limit and universality
AU - Jacopo Borga
AU - Ewain Gwynne
AU - Yuanzheng Wang
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22514
UR - https://arxiv.org/abs/2609.22514
AB - We define a natural orientation on a spanning-tree-decorated planar map whereby, roughly speaking, each directed edge in the map is oriented to match the direction of the contour exploration of the spanning tree. We study directed distances (lengths of shortest directed paths) with respect to this orientation. We construct the Busemann function which measures directed distances to ∞ along a natural interface in the uniform infinite spanning-tree-decorated map. We show that this Busemann function, re-scaled appropriately, converges in law to a 3/2-stable Lévy process. We also show that in a uniform spanning-tree-decorated map with n edges, directed distances are typically of order n^1/3. Using a strong coupling argument, we deduce analogous statements for directed distances in other random planar maps in the √2-Liouville quantum gravity (LQG) universality class, including uniform meandric systems and mated-CRT maps for γ=√2. These results give the scaling dimension for a hypothetical directed version of the √2-LQG metric. Our proof strategy is inspired by work of Borga and Gwynne (2025) on directed distances in bipolar-oriented triangulations.
ER -