A distance exponent for Liouville quantum gravity

Ewain Gwynne, Nina Holden, Xin Sun

Published 2016-06-03Version 1

Let $\gamma \in (0,2)$ and let $h$ be a random distribution on $\mathbb C$ which parametrizes 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 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 $(h, \eta)$, which we call the LQG structure graphs, and which we conjecture converges in the scaling limit in the Gromov-Hausdorff topology to a metric on the $\gamma$-LQG cone. In particular, for $\mathcal G^\epsilon$ is the graph whose vertices are segments of the form $\eta([(k-1)\epsilon , k\epsilon])$ for $k\in\mathbb Z$, with two such segments connected by an edge if and only if they 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 certain correlated two-dimensional Brownian motion, so can be studied without any reference to SLE or LQG. We 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 $\epsilon^{-\chi + o_\epsilon(1)}$. In the special case when $\gamma = \sqrt 2$, we show that the diameter of this subgraph of $\mathcal G^\epsilon$ is of order $\epsilon^{-\chi + o_\epsilon(1)}$ with high probability. We also 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.