Equivalence of metric gluing and conformal welding in $γ$-Liouville quantum gravity for $γ\in (0,2)$
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arXiv
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| Format: | Preprint |
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2022
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| author | Hughes, Liam Miller, Jason |
| author_facet | Hughes, Liam Miller, Jason |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_00589 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Equivalence of metric gluing and conformal welding in $γ$-Liouville quantum gravity for $γ\in (0,2)$ Hughes, Liam Miller, Jason Probability Mathematical Physics Metric Geometry 60D05 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. |
| title | Equivalence of metric gluing and conformal welding in $γ$-Liouville quantum gravity for $γ\in (0,2)$ |
| topic | Probability Mathematical Physics Metric Geometry 60D05 |
| url | https://arxiv.org/abs/2212.00589 |