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

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 b