2020/10/15 by Ewain Gwynne, Gwynne, Ewain, Joshua Pfeffer +4
Mathematics · Physics and Astronomy · #Advanced Topology and Set Theory #Black Holes and Theoretical Physics #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Noncommutative and Quantum Gravity Theories #Probability (math.PR) #math-ph #math.MP #math.PR
paper · pdf · doi:10.48550/arxiv.2010.07889
44 pages, 9 figures, to appear in PTRF
openalex publication_date 2020/10/15 · arxiv created 2022/01/17 · arxiv updated 2022/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Recent works have shown that there is a canonical way to to assign a metric (distance function) to a Liouville quantum gravity (LQG) surface for any parameter γ∈ (0,2). We establish a strong confluence property for LQG geodesics, which generalizes a result proven by Angel, Kolesnik and Miermont for the Brownian map. Using this property, we also establish zero-one laws for the Hausdorff dimensions of geodesics, metric ball boundaries, and metric nets w.r.t. the Euclidean or LQG metric. In the case of a metric ball boundary, our result combined with earlier work of Gwynne (2020) gives a formula for the a.s. Hausdorff dimension for the boundary of the metric ball stopped when it hits a fixed point in terms of the Hausdorff dimension of the whole LQG surface. We also show that the Hausdorff dimension of the metric ball boundary is carried by points which are not on the boundary of any complementary connected component of the ball.