CONFLUENCE OF GEODESICS IN LIOUVILLE QUANTUM GRAVITY FOR γ ∈ (0,2)

• 成果类型: Article
• 署名作者: Gwynne, Ewain; Miller, Jason
• 署名单位: University of Cambridge
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1409
• 发表日期: 2020
• 页码: 1861-1901
• 关键词: geometry

摘要

We prove that for any metric, which one can associate with a Liouville quantum gravity (LQG) surface for gamma is an element of (0, 2) satisfying certain natural axioms, its geodesics exhibit the following confluence property. For any fixed point z, a.s. any two gamma-LQG geodesics started from distinct points other than z must merge into each other and subsequently coincide until they reach z. This is analogous to the confluence of geodesics property for the Brownian map proven by Le Gall (Acta Math. 205 (2010) 287-360). Our results apply for the subsequential limits of Liouville first passage percolation and are an important input in the proof of the existence and uniqueness of the LQG metric for all gamma is an element of (0, 2).