The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity
Vortrag von Ewain Gwynne
Datum: 11.10.17 Zeit: 17.15 - 18.15 Raum: ETH HG G 43
We discuss the first proof that the discrete conformal embeddings of certain random planar maps approximate their continuum counterparts. In particular, we show that the Tutte embeddings (a.k.a. harmonic or barycentric embeddings) of the mated-CRT maps maps converge to $\gamma$-Liouville quantum gravity (LQG). Mated-CRT maps 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 (in the annealed sense) and that the embedded random walk converges to Brownian motion (in the quenched sense).
Our proof proceeds by way of a quenched scaling limit result for random walk in a certain inhomogeneous random environment. 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 https://arxiv.org/abs/1705.11161.