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arXiv · 2312.12089

Liouville quantum gravity metrics are not doubling

Liam Hughes
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Abstract

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 $γ= \sqrt{8/3}$).

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Liam Hughes. 2024-08-01. Liouville quantum gravity metrics are not doubling. https://doi.org/10.1214/24-ecp607

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