CROSSOVER DISTRIBUTIONS AT THE EDGE OF THE RAREFACTION FAN

• 成果类型: Article
• 署名作者: Corwin, Ivan; Quastel, Jeremy
• 署名单位: New York University; University of Toronto
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP725
• 发表日期: 2013
• 页码: 1243-1314
• 关键词: polynuclear growth-model; current fluctuations; equation; limit; covariance; tasep; asep

摘要

We consider the weakly asymmetric limit of simple exclusion process with drift to the left, starting from step Bernoulli initial data with rho(-) < rho(+) so that macroscopically one has a rarefaction fan. We study the fluctuations of the process observed along slopes in the fan, which are given by the Hopf-Cole solution of the Kardar-Parisi-Zhang (KPZ) equation, with appropriate initial data. For slopes strictly inside the fan, the initial data is a Dirac delta function and the one point distribution functions have been computed in [Comm. Pure Appl. Math. 64 (2011) 466-537] and [Nuclear Phys. B 834 (2010) 523-542]. At the edge of the rarefaction fan, the initial data is one-sided Brownian. We obtain a new family of crossover distributions giving the exact one-point distributions of this process, which converge, as T NE arrow infinity to those of the Airy A(2 -> BM) process. As an application, we prove moment and large deviation estimates for the equilibrium Hopf-Cole solution of KPZ. These bounds rely on the apparently new observation that the FKG inequality holds for the stochastic heat equation. Finally, via a Feynman-Kac path integral, the KPZ equation also governs the free energy of the continuum directed polymer, and thus our formula may also be interpreted in those terms.