KPZ exponents for the half-space log-gamma polymer

• 成果类型: Article
• 署名作者: Barraquand, Guillaume; Corwin, Ivan; Das, Sayan
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite PSL; Ecole Normale Superieure (ENS); Universite Paris Cite; Sorbonne Universite; Columbia University; University of Chicago
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01324-x
• 发表日期: 2025
• 页码: 1113-1243
• 关键词: CENTRAL LIMIT-THEOREMS; schur process; directed polymers; MODEL; point; asep; fluctuations; TRANSITION; continuity; diffusion

摘要

We consider the point-to-point log-gamma polymer of length 2N in a half-space with i.i.d. Gamma(-1)(2 theta) distributed bulk weights and i.i.d. Gamma(-1)(alpha + theta) distributed boundary weights for theta > 0 and alpha > -theta. We establish the KPZ exponents (1/3 fluctuation and 2/3 transversal) for this model when alpha = N-1/3 mu for mu is an element of R fixed (critical regime) and when alpha > 0 is fixed (supercritical regime). In particular, in these two regimes, we show that after appropriate centering, the free energy process with spatial coordinate scaled by N-2/3 and fluctuations scaled by N-1/3 is tight. These regimes correspond to a polymer measure which is not pinned at the boundary. This is the first instance of establishing the 2/3 transversal exponent for a positive temperature half-space model, and the first instance of the 1/3 fluctuation exponent besides precisely at the boundary where recent work of Imamura et al. (Solvable models in the KPZ class: approach through periodic and free boundary Schur measures. arXiv:2204.08420. 2022) applies and also gives the exact one-point fluctuation distribution (our methods do not access exact fluctuation distributions). Our proof relies on two inputs-the relationship between the half-space log-gamma polymer and half-space Whittaker process (facilitated by the geometric RSK correspondence as initiated in Corwin et al. (Duke Math J 163(3):513-563, 2014), O'Connell et al. (Invent Math 197(2):361-416, 2014), and an identity in Barraquand and Wang (Int Math Res Not 2023:11877, 2022) which relates the point-to-line half-space partition function to the full-space partition function for the log-gamma polymer. The primary technical contribution of our work is to construct the half-space log-gamma Gibbsian line ensemble and develop, in the spirit of work initiated in Corwin and Hammond (Invent Math 195(2):441-508, 2014), a toolbox for extracting tightness and absolute continuity results from minimal information about the top curve of such half-space line ensembles. This is the first study of half-space line ensembles.