KP governs random growth off a 1-dimensional substrate. (21st April 2022)

• Record Type: Journal Article
• Authors: Quastel, Jeremy; Remenik, Daniel
• Is Part Of: Forum of mathematics. Volume 10(2022)
• Publication Date: 2022-04-21
• Subjects: 60K35 -- 82C22; Mathematics -- Periodicals; 510
• DOI: 10.1017/fmp.2021.9
• Languages: English
• ISSNs: 2050-5086

Abstract

Abstract: The logarithmic derivative of the marginal distributions of randomly fluctuating interfaces in one dimension on a large scale evolve according to the Kadomtsev–Petviashvili (KP) equation. This is derived algebraically from a Fredholm determinant obtained in [MQR17] for the Kardar–Parisi–Zhang (KPZ) fixed point as the limit of the transition probabilities of TASEP, a special solvable model in the KPZ universality class. The Tracy–Widom distributions appear as special self-similar solutions of the KP and Korteweg–de Vries equations. In addition, it is noted that several known exact solutions of the KPZ equation also solve the KP equation.