A distance exponent for Liouville quantum gravity

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 \textit{LQG structure graphs} (a.k.a.\ \textit{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, Miller, and Sheffield (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 (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 $\epsilon^{-\chi + o_\epsilon(1)}$.