A distance exponent for Liouville quantum gravity

Let γ∈ (0 , 2) and let h be the random distribution on C which describes a γ-Liouville quantum gravity (LQG) cone. Also let κ= 16 / γ ² > 4 and let η be a whole-plane space-filling SLE _κ curve sampled independent from h and parametrized by γ-quantum mass with respect to h. We study a family {Gϵ}ϵ>0 of planar maps associated with (h, η) 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 γ-LQG. In particular, G ^ϵ is the graph whose vertex set is ϵZ, with two such vertices x ₁ , x ₂ ∈ ϵZ connected by an edge if and only if the corresponding curve segments η([x ₁ - ϵ, x ₁ ]) and η([x ₂ - ϵ, 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 G ^ϵ 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 G ^ϵ 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 χ> 0 for which the expected graph distance between generic points in the subgraph of G ^ϵ corresponding to the segment η([0 , 1]) is of order ϵ-χ+oϵ(1), and this distance is extremely unlikely to be larger than ϵ-χ+oϵ(1).