The Tutte embedding of tree-weighted planar maps converges to Liouville quantum gravity

Nina Holden

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arXiv · 2610.10501

The Tutte embedding of tree-weighted planar maps converges to Liouville quantum gravity

Nina Holden

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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 $γ=\sqrt{2}$ 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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Nina Holden. 2026-10-07. The Tutte embedding of tree-weighted planar maps converges to Liouville quantum gravity. https://arxiv.org/abs/2610.10501

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