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The Tutte embedding of tree-weighted planar maps converges to Liouville quantum gravity
Abstract
Suppose M is a planar map with a macroscopic boundary decorated by a spanning tree T, such that (M,T) has been sampled from the critical Boltzman measure. We show that M converges to a Liouville quantum gravity disk with parameter γ=√2 under the Tutte embedding as the length of the boundary of M goes to infinity. Furthermore, conditioned on (M,T), the simple random walk on M converges to a planar Brownian motion modulo reparametrization of time.
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BibTeX
@misc{holden2026tutte,
title = {{The Tutte embedding of tree-weighted planar maps converges to Liouville quantum gravity}},
author = {Nina Holden},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10501},
eprint = {2610.10501},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10501},
}
RIS
TY - UNPB
TI - The Tutte embedding of tree-weighted planar maps converges to Liouville quantum gravity
AU - Nina Holden
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10501
UR - https://arxiv.org/abs/2610.10501
AB - Suppose M is a planar map with a macroscopic boundary decorated by a spanning tree T, such that (M,T) has been sampled from the critical Boltzman measure. We show that M converges to a Liouville quantum gravity disk with parameter γ=√2 under the Tutte embedding as the length of the boundary of M goes to infinity. Furthermore, conditioned on (M,T), the simple random walk on M converges to a planar Brownian motion modulo reparametrization of time.
ER -