Geodesics and metric ball boundaries in Liouville quantum gravity
Abstract Recent works have shown that there is a canonical way to to assign a metric (distance function) to a Liouville quantum gravity (LQG) surface for any parameter $$\gamma \in (0,2)$$...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer Berlin Heidelberg
2022
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| Online Access: | https://hdl.handle.net/1721.1/141926 |
| _version_ | 1826194387324895232 |
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| author | Gwynne, Ewain Pfeffer, Joshua Sheffield, Scott |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Gwynne, Ewain Pfeffer, Joshua Sheffield, Scott |
| author_sort | Gwynne, Ewain |
| collection | MIT |
| description | Abstract
Recent works have shown that there is a canonical way to to assign a metric (distance function) to a Liouville quantum gravity (LQG) surface for any parameter
$$\gamma \in (0,2)$$
γ
∈
(
0
,
2
)
. We establish a strong confluence property for LQG geodesics, which generalizes a result proven by Angel, Kolesnik and Miermont for the Brownian map. Using this property, we also establish zero-one laws for the Hausdorff dimensions of geodesics, metric ball boundaries, and metric nets w.r.t. the Euclidean or LQG metric. In the case of a metric ball boundary, our result combined with earlier work of Gwynne (Commun Math Phys 378(1):625–689, 2020.
arXiv:1909.08588
) gives a formula for the a.s. Hausdorff dimension for the boundary of the metric ball stopped when it hits a fixed point in terms of the Hausdorff dimension of the whole LQG surface. We also show that the Hausdorff dimension of the metric ball boundary is carried by points which are not on the boundary of any complementary connected component of the ball. |
| first_indexed | 2024-09-23T09:55:19Z |
| format | Article |
| id | mit-1721.1/141926 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T09:55:19Z |
| publishDate | 2022 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1419262023-12-06T21:12:11Z Geodesics and metric ball boundaries in Liouville quantum gravity Gwynne, Ewain Pfeffer, Joshua Sheffield, Scott Massachusetts Institute of Technology. Department of Mathematics Abstract Recent works have shown that there is a canonical way to to assign a metric (distance function) to a Liouville quantum gravity (LQG) surface for any parameter $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) . We establish a strong confluence property for LQG geodesics, which generalizes a result proven by Angel, Kolesnik and Miermont for the Brownian map. Using this property, we also establish zero-one laws for the Hausdorff dimensions of geodesics, metric ball boundaries, and metric nets w.r.t. the Euclidean or LQG metric. In the case of a metric ball boundary, our result combined with earlier work of Gwynne (Commun Math Phys 378(1):625–689, 2020. arXiv:1909.08588 ) gives a formula for the a.s. Hausdorff dimension for the boundary of the metric ball stopped when it hits a fixed point in terms of the Hausdorff dimension of the whole LQG surface. We also show that the Hausdorff dimension of the metric ball boundary is carried by points which are not on the boundary of any complementary connected component of the ball. 2022-04-19T15:05:35Z 2022-04-19T15:05:35Z 2022-02-11 2022-04-17T03:37:26Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/141926 Gwynne, Ewain, Pfeffer, Joshua and Sheffield, Scott. 2022. "Geodesics and metric ball boundaries in Liouville quantum gravity." en https://doi.org/10.1007/s00440-022-01112-5 Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature application/pdf Springer Berlin Heidelberg Springer Berlin Heidelberg |
| spellingShingle | Gwynne, Ewain Pfeffer, Joshua Sheffield, Scott Geodesics and metric ball boundaries in Liouville quantum gravity |
| title | Geodesics and metric ball boundaries in Liouville quantum gravity |
| title_full | Geodesics and metric ball boundaries in Liouville quantum gravity |
| title_fullStr | Geodesics and metric ball boundaries in Liouville quantum gravity |
| title_full_unstemmed | Geodesics and metric ball boundaries in Liouville quantum gravity |
| title_short | Geodesics and metric ball boundaries in Liouville quantum gravity |
| title_sort | geodesics and metric ball boundaries in liouville quantum gravity |
| url | https://hdl.handle.net/1721.1/141926 |
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