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

-
Ewain Gwynne , MIT
Fine Hall 110

We discuss the proof that the Tutte embeddings (a.k.a. harmonic or barycentric 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 sense) and that the embedded random walk converges to Brownian motion (in the quenched sense). Several recent papers have shown that random planar maps converge to SLE-decorated LQG in important ways (as path-decorated metric spaces, as mated pairs of trees, etc.) but this is the first result which shows that discrete conformal embeddings of random planar maps approximate their continuum counterparts.  The mated-CRT map provides a coarse-grained approximation of other random planar maps which can be bijectively encoded by pairs of discrete random trees---e.g., the UIPT, spanning-tree weighted maps, and bipolar-oriented maps---so our results suggest a possible approach for proving that embeddings of these planar maps also converge to LQG. Based on joint work with Jason Miller and Scott Sheffield