# The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity

@article{Gwynne2017TheTE, title={The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity}, author={Ewain Gwynne and Jason Miller and Scott Sheffield}, journal={The Annals of Probability}, year={2017} }

We prove that the Tutte embeddings (a.k.a. harmonic/embeddings) of certain random planar maps converge to $\gamma$-Liouville quantum gravity ($\gamma$-LQG). Specifically, we treat mated-CRT maps, which are discretized matings of correlated continuum random trees, and $\gamma$ ranges from $0$ to $2$ as one varies the correlation parameter. We also show that the associated space-filling path on the embedded map converges to space-filling SLE$_{\kappa}$ for $\kappa =16/\gamma^2$ (in the annealed…

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The peanosphere construction of Duplantier, Miller, and Sheffield provides a means of representing a $\gamma$-Liouville quantum gravity (LQG) surface, $\gamma \in (0,2)$, decorated with a…

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