Random-walk in Beta-distributed random environment

• 成果类型: Article
• 署名作者: Barraquand, Guillaume; Corwin, Ivan
• 署名单位: Columbia University; Sorbonne Universite; Universite Paris Cite; Sorbonne Universite
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-016-0699-z
• 发表日期: 2017
• 页码: 1057-1116
• 关键词: free-energy fluctuations; tracy-widom asymptotics; central-limit-theorem; directed polymer

摘要

We introduce an exactly-solvable model of random walk in random environment that we call the Beta RWRE. This is a random walk in which performs nearest neighbour jumps with transition probabilities drawn according to the Beta distribution. We also describe a related directed polymer model, which is a limit of the q-Hahn interacting particle system. Using a Fredholm determinant representation for the quenched probability distribution function of the walker's position, we are able to prove second order cube-root scale corrections to the large deviation principle satisfied by the walker's position, with convergence to the Tracy-Widom distribution. We also show that this limit theorem can be interpreted in terms of the maximum of strongly correlated random variables: the positions of independent walkers in the same environment. The zero-temperature counterpart of the Beta RWRE can be studied in a parallel way. We also prove a Tracy-Widom limit theorem for this model.