INTEGRATION BY PARTS AND THE KPZ TWO-POINT FUNCTION

• 成果类型: Article
• 署名作者: Pimentel, Leandro P. R.
• 署名单位: Universidade Federal do Rio de Janeiro
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1564
• 发表日期: 2022
• 页码: 1755-1780
• 关键词: polynuclear growth; directed polymers; FIXED-POINT; distributions; fluctuations; tasep; limit; MODEL

摘要

In this article, we consider the KPZ fixed point starting from a two-sided Brownian motion with an arbitrary diffusion coefficient. We apply the integration by parts formula from Malliavin calculus to establish a key relation between the two-point (correlation) function of the spatial derivative process and the location of the maximum of an Airy process plus Brownian motion minus a parabola. Integration by parts also allows us to deduce the density of this location in terms of the second derivative of the variance of the KPZ fixed point. In the stationary regime, we find the same density related to limit fluctuations of a second-class particle. We further develop an adaptation of Stein's method that implies asymptotic independence of the spatial derivative process from the initial data.