2016/06/03 by Ewain Gwynne, Gwynne, Ewain, Nina Holden +3
Computer Science · Mathematics · #Data Management and Algorithms #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #Mathematical Physics (math-ph) #Probability (math.PR) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1606.01214
openalex publication_date 2016/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let γ∈ (0,2) and let h be the random distribution on \mathbb C which describes a γ-Liouville quantum gravity (LQG) cone. Also let κ= 16/γ2 >4 and let η be a whole-plane space-filling SLEκ curve sampled independent from h and parametrized by γ-quantum mass with respect to h. We study a family \\mathcal 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, \mathcal Gε is the graph whose vertex set is ε\mathbb Z, with two such vertices x1,x2∈ ε\mathbb Z connected by an edge if and only if the corresponding curve segments η([x1-ε, x1]) and η([x2-ε,x2]) 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 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 Gε 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 χ> 0 for which the expected graph distance between generic points in the subgraph of \mathcal Gε 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).