ENTROPY-CONTROLLED LAST-PASSAGE PERCOLATION

• 成果类型: Article
• 署名作者: Berger, Quentin; Torri, Niccolo
• 署名单位: Sorbonne Universite
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/18-AAP1448
• 发表日期: 2019
• 页码: 1878-1903
• 关键词: directed polymers

摘要

We introduce a natural generalization of Hammersley's Last-Passage Percolation (LPP) called Entropy-controlled Last-Passage Percolation (E-LPP), where points can be collected by paths with a global (path-entropy) constraint which takes into account the whole structure of the path, instead of a local (1-Lipschitz) constraint as in Hammersley's LPP. Our main result is to prove quantitative tail estimates on the maximal number of points that can be collected by a path with entropy bounded by a prescribed constant. The E-LPP turns out to be a key ingredient in the context of the directed polymer model when the environment is heavy-tailed, which we consider in (Berger and Toni (2018)). We give applications in this context, which are essentials tools in (Berger and Toni (2018)): we show that the limiting variational problem conjectured in (Ann. Probab. 44 (2016) 4006-4048), Conjecture 1.7, is finite, and we prove that the discrete variational problem converges to the continuous one, generalizing techniques used in (Comm. Pure Appl. Math. 64 (2011) 183-204; Probab. Theory Related Fields 137 (2007) 227-275).