Liouville quantum gravity metrics are not doubling

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 $\gamma \in (0,2]$, the $\gamma$-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 $\gamma =\sqrt{8/3}$).

12 pages, 1 figure

References in corpus (1)

• The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds