• DOI:10.1007/s00440-018-0846-9
• Corpus ID: 54674783

A distance exponent for Liouville quantum gravity

@article{Gwynne2016ADE,
  title={A distance exponent for Liouville quantum gravity},
  author={Ewain Gwynne and Nina Holden and Xin Sun},
  journal={Probability Theory and Related Fields},
  year={2016},
  volume={173},
  pages={931-997}
}

• Ewain Gwynne, N. Holden, Xin Sun
• Published 3 June 2016
• Mathematics
• Probability Theory and Related Fields

Let $$\gamma \in (0,2)$$γ∈(0,2) and let h be the random distribution on $$\mathbb C$$C which describes a $$\gamma $$γ-Liouville quantum gravity (LQG) cone. Also let $$\kappa = 16/\gamma ^2 >4$$κ=16/γ2>4 and let $$\eta $$η be a whole-plane space-filling SLE$$_\kappa $$κ curve sampled independent from h and parametrized by $$\gamma $$γ-quantum mass with respect to h. We study a family $$\{\mathcal G^\epsilon \}_{\epsilon >0}$${Gϵ}ϵ>0 of planar maps associated with $$(h, \eta )$$(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 $$\gamma $$γ-LQG. In particular, $$\mathcal G^\epsilon $$Gϵ is the graph whose vertex set is $$\epsilon \mathbb Z$$ϵZ, with two such vertices $$x_1,x_2\in \epsilon \mathbb Z$$x1,x2∈ϵZ connected by an edge if and only if the corresponding curve segments $$\eta ([x_1-\epsilon , x_1])$$η([x1-ϵ,x1]) and $$\eta ([x_2-\epsilon ,x_2])$$η([x2-ϵ,x2]) 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 $$\mathcal G^\epsilon $$Gϵ 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 $$\mathcal G^\epsilon $$Gϵ 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 $$\chi > 0$$χ>0 for which the expected graph distance between generic points in the subgraph of $$\mathcal G^\epsilon $$Gϵ corresponding to the segment $$\eta ([0,1])$$η([0,1]) is of order $$\epsilon ^{-\chi + o_\epsilon (1)}$$ϵ-χ+oϵ(1), and this distance is extremely unlikely to be larger than $$\epsilon ^{-\chi + o_\epsilon (1)}$$ϵ-χ+oϵ(1).