KPZ line ensemble

• 成果类型: Article
• 署名作者: Corwin, Ivan; Hammond, Alan
• 署名单位: Columbia University; Sorbonne Universite; University of California System; University of California Berkeley; University of California System; University of California Berkeley
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0651-7
• 发表日期: 2016
• 页码: 67-185
• 关键词: dimensional directed polymer; stochastic heat-equation; brownian-motion; free-energy; fluctuations

摘要

For each t >= 1we construct an -indexed ensemble of random continuous curves with three properties: the lowest indexed curve is distributed as the time t Hopf-Cole solution to the Kardar-Parisi-Zhang (KPZ) stochastic partial differential equation with narrow wedge initial data; the entire ensemble satisfies a resampling invariance which we call the -Brownian Gibbs property [with H(x) = e(x)]; increments of the lowest indexed curve, when centered by and scaled down vertically by and horizontally by , remain uniformly absolutely continuous (i.e. have tight Radon-Nikodym derivatives) with respect to Brownian bridges as time t goes to infinity. This construction uses as inputs the diffusion that O'Connell discovered (Ann Probab 40:437-458, 2012) in relation to the O'Connell-Yor semi-discrete Brownian polymer, the convergence result of Moreno Flores et al. (in preparation) of the lowest indexed curve of that diffusion to the solution of the KPZ equation with narrow wedge initial data, and the one-point distribution formula proved by Amir et al. (Commun Pure Appl Math 64:466-537, 2011) for the solution of the KPZ equation with narrow wedge initial data. We provide four main applications of this construction: uniform (as t goes to infinity) Brownian absolute continuity of the time t solution to the KPZ equation with narrow wedge initial data, even when scaled vertically by and horizontally by ; universality of the one-point (vertical) fluctuation scale for the solution of the KPZ equation with general initial data; concentration in the scale for the endpoint of the continuum directed random polymer; exponential upper and lower tail bounds for the solution at fixed time of the KPZ equation with general initial data.