2018/12/31 by Jian Ding, Alexander Dunlap · 1 citation
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Combinatorics #Distance #Euclidean distance #Euclidean geometry #Geometry #Graph #Markov Chains and Monte Carlo Methods #Mathematics #Noncommutative and Quantum Gravity Theories #Physics #Quantum #Quantum gravity #Quantum mechanics #Scaling #Shortest path problem #math-ph #math.MP #math.PR #msc:60G60 #msc:60K35
paper · pdf · doi:10.1007/s00220-020-03684-6
published as Comm. Math. Phys., Volume 376 (2020), 1499-1572 · 65 pages, 10 figures
arxiv created 2019/10/03 · openalex publication_date 2020/03/11 · arxiv updated 2020/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
For 0<γ<2 and δ>0, we consider the Liouville graph distance, which is the minimal number of Euclidean balls of γ-Liouville quantum gravity measure at most δ whose union contains a continuous path between two endpoints. In this paper, we show that the renormalized distance is tight and thus has subsequential scaling limits at δ→ 0. In particular, we show that for all δ>0 the diameter with respect to the Liouville graph distance has the same order as the typical distance between two endpoints.