Liouville quantum gravity metrics are not doubling
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929445526306816 |
|---|---|
| author | Hughes, Liam |
| author_facet | Hughes, Liam |
| contents | 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 $γ= \sqrt{8/3}$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_12089 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Liouville quantum gravity metrics are not doubling Hughes, Liam Probability Metric Geometry 60D05 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 $γ= \sqrt{8/3}$). |
| title | Liouville quantum gravity metrics are not doubling |
| topic | Probability Metric Geometry 60D05 |
| url | https://arxiv.org/abs/2312.12089 |