Geodesics and metric ball boundaries in Liouville quantum gravity

• 成果类型: Article
• 署名作者: Gwynne, Ewain; Pfeffer, Joshua; Sheffield, Scott
• 署名单位: University of Chicago; Columbia University; Massachusetts Institute of Technology (MIT)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-022-01112-5
• 发表日期: 2022
• 页码: 905-954
• 关键词: fractal structure; brownian map; sle

摘要

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 gamma is an element of(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 (Commun Math Phys 378(1):625-689, 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.