RATIOS OF PARTITION FUNCTIONS FOR THE LOG-GAMMA POLYMER

• 成果类型: Article
• 署名作者: Georgiou, Nicos; Rassoul-Agha, Firas; Seppaelaeinen, Timo; Yilmaz, Atilla
• 署名单位: University of Sussex; Utah System of Higher Education; University of Utah; University of Wisconsin System; University of Wisconsin Madison; Bogazici University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP933
• 发表日期: 2015
• 页码: 2282-2331
• 关键词: random environment; competition interfaces; whittaker functions; directed polymers; random potentials; random-walks; free-energy; disorder

摘要

We introduce a random walk in random environment associated to an underlying directed polymer model in 1 + 1 dimensions. This walk is the positive temperature counterpart of the competition interface of percolation and arises as the limit of quenched polymer measures. We prove this limit for the exactly solvable log-gamma polymer, as a consequence of almost sure limits of ratios of partition functions. These limits of ratios give the Busemann functions of the log-gamma polymer, and furnish centered cocycles that solve a variational formula for the limiting free energy. Limits of ratios of point-to-point and point-to-line partition functions manifest a duality between tilt and velocity that comes from quenched large deviations under polymer measures. In the log-gamma case, we identify a family of ergodic invariant distributions for the random walk in random environment.