BIPOLAR ORIENTATIONS ON PLANAR MAPS AND SLE12

• 成果类型: Article
• 署名作者: Kenyon, Richard; Miller, Jason; Sheffield, Scott; Wilson, David B.
• 署名单位: Brown University; University of Cambridge; Massachusetts Institute of Technology (MIT); University of Washington; University of Washington Seattle
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1282
• 发表日期: 2019
• 页码: 1240-1269
• 关键词: brownian intersection exponents; erased random-walks; QUANTUM-GRAVITY; conformal-invariance; VALUES; reversibility; Duality; trees

摘要

We give bijections between bipolar-oriented (acyclic with unique source and sink) planar maps and certain random walks, which show that the uniformly random bipolar-oriented planar map, decorated by the peano curve surrounding the tree of left-most paths to the sink, converges in law with respect to the peanosphere topology to a root 4/3-Liouville quantum gravity surface decorated by an independent Schramm-Loewner evolution with parameter kappa = 12 (i.e., SLE12). This result is universal in the sense that it holds for bipolar-oriented triangulations, quadrangulations, k-angulations and maps in which face sizes are mixed.