# A distance exponent for Liouville quantum gravity

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## Abstract

Let \(\gamma \in (0,2)\) and let *h* be the random distribution on \(\mathbb C\) which describes a \(\gamma \)-Liouville quantum gravity (LQG) cone. Also let \(\kappa = 16/\gamma ^2 >4\) and let \(\eta \) be a whole-plane space-filling SLE\(_\kappa \) curve sampled independent from *h* and parametrized by \(\gamma \)-quantum mass with respect to *h*. We study a family \(\{\mathcal G^\epsilon \}_{\epsilon >0}\) of planar maps associated with \((h, \eta )\) called the *LQG structure graphs* (a.k.a. *mated-CRT maps*) which we conjecture converge in probability in the scaling limit with respect to the Gromov–Hausdorff topology to a random metric space associated with \(\gamma \)-LQG. In particular, \(\mathcal G^\epsilon \) is the graph whose vertex set is \(\epsilon \mathbb Z\), with two such vertices \(x_1,x_2\in \epsilon \mathbb Z\) connected by an edge if and only if the corresponding curve segments \(\eta ([x_1-\epsilon , x_1])\) and \(\eta ([x_2-\epsilon ,x_2])\) share a non-trivial boundary arc. Due to the peanosphere description of SLE-decorated LQG due to Duplantier et al. (Liouville quantum gravity as a mating of trees, 2014), the graph \(\mathcal G^\epsilon \) can equivalently be expressed as an explicit functional of a correlated two-dimensional Brownian motion, so can be studied without any reference to SLE or LQG. We prove non-trivial upper and lower bounds for the cardinality of a graph-distance ball of radius *n* in \(\mathcal G^\epsilon \) which are consistent with the prediction of Watabiki (Prog Theor Phys Suppl 114:1–17, 1993) for the Hausdorff dimension of LQG. Using subadditivity arguments, we also prove that there is an exponent \(\chi > 0\) for which the expected graph distance between generic points in the subgraph of \(\mathcal G^\epsilon \) corresponding to the segment \(\eta ([0,1])\) is of order \(\epsilon ^{-\chi + o_\epsilon (1)}\), and this distance is extremely unlikely to be larger than \(\epsilon ^{-\chi + o_\epsilon (1)}\).

## Mathematics Subject Classification

60J67 (SLE) 60D05 (geometric probability) 60J65 (Brownian motion)## Notes

### Acknowledgements

We thank Jian Ding, Subhajit Goswami, Jason Miller, and Scott Sheffield for helpful discussions. E.G. was supported by the U.S. Department of Defense via an NDSEG fellowship. N.H. was supported by a doctoral research fellowship from the Norwegian Research Council. X.S. was supported by the Simons Foundation as a Junior Fellow at Simons Society of Fellows. We thank two anonymous referees for helpful comments on an earlier version of this article.

## Supplementary material

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