A distance exponent for Liouville quantum gravity

• 成果类型: Article
• 署名作者: Gwynne, Ewain; Holden, Nina; Sun, Xin
• 署名单位: Massachusetts Institute of Technology (MIT); Columbia University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-018-0846-9
• 发表日期: 2019
• 页码: 931-997
• 关键词: 2d

摘要

Let (0,2) and let h be the random distribution on C which describes a -Liouville quantum gravity (LQG) cone. Also let =16/2>4 and let be a whole-plane space-filling SLE curve sampled independent from h and parametrized by -quantum mass with respect to h. We study a family {GE}E>0 of planar maps associated with (h,) 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 -LQG. In particular, GE is the graph whose vertex set is EZ, with two such vertices x1,x2EZ connected by an edge if and only if the corresponding curve segments ([x1-E,x1]) and ([x2-E,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 GE 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 GE 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 >0 for which the expected graph distance between generic points in the subgraph of GE corresponding to the segment ([0,1]) is of order E-+oE(1), and this distance is extremely unlikely to be larger than E-+oE(1).