THREE-HALVES VARIATION OF GEODESICS IN THE DIRECTED LANDSCAPE

• 成果类型: Article
• 署名作者: Dauvergne, Duncan; Sarkar, Sourav; Virag, Balint
• 署名单位: University of Toronto; University of Cambridge
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1574
• 发表日期: 2022
• 页码: 1947-1985
• 关键词: fluctuations; continuity

摘要

We show that geodesics in the directed landscape have 3/2-variation and that weight functions along the geodesics have cubic variation. We show that the geodesic and its landscape environment around an interior point have a small-scale limit. This limit is given in terms of the directed landscape with Brownian-Bessel boundary conditions. The environments around different interior points are asymptotically independent. We give tail bounds with optimal exponents for geodesic and weight function increments. As an application of our results, we show that geodesics are not Holder-2/3 and that weight functions are not Holder-1/3, although these objects are known to be Holder with all lower exponents.