Liam Hughes | Searcharxiv

arXiv2024

Liouville quantum gravity metrics are not doubling

We observe that non-doubling metric spaces can be characterized as those that contain arbitrarily large sets of approximately equidistant points and use this to show that, for $γ\in (0,2]$, the $γ$-Liouville quantum gravity metric is almost surely not doubling and thus cannot be quasisymmetrically embedded into any finite-dimensional Euclidean space. This generalizes the corresponding result of Troscheit for the Brownian map (which is equivalent to the case $γ= \sqrt{8/3}$).

math.PR↗

arXiv2024

Equivalence of metric gluing and conformal welding in $γ$-Liouville quantum gravity for $γ\in (0,2)$

We consider the $γ$-Liouville quantum gravity (LQG) model for $γ\in (0,2)$, formally described by $e^{γh}$ where $h$ is a Gaussian free field on a planar domain $D$. Sheffield showed that when a certain type of LQG surface, called a quantum wedge, is decorated by an appropriate independent SLE curve, the wedge is cut into two independent surfaces which are themselves quantum wedges, and that the original surface can be recovered as a unique conformal welding. We prove that the original surface can also be obtained as a metric space quotient of the two wedges, extending results of Gwynne and Miller in the special case $γ= \sqrt{8/3}$ to the whole subcritical regime $γ\in (0,2)$. Since the proof for $γ= \sqrt{8/3}$ used estimates for Brownian surfaces, which are equivalent to $γ$-LQG surfaces only when $γ=\sqrt{8/3}$, we instead use GFF techniques to establish estimates relating distances, areas and boundary lengths, as well as bi-Hölder continuity of the LQG metric w.r.t. the Euclidean metric at the boundary, which may be of independent interest.

math.PR↗