KP governs random growth off a one dimensional substrate

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, arXiv:1701.00018] for the 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 KP and KdV. In addition, it is noted that several known exact solutions of the KPZ equation also solve KP.

Improved presentation, misprint corrected. 22 pages

References in corpus (6)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Fluctuation properties of the TASEP with periodic initial configuration
• The KPZ equation with flat initial condition and the directed polymer with one free end
• Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function
• PDEs for the joint distributions of the Dyson, Airy and Sine processes
• Riemann surfaces for KPZ with periodic boundaries

Cited by in corpus (7)

• Exact solution of the macroscopic fluctuation theory for the symmetric exclusion process
• Uniform tail asymptotics for Airy kernel determinant solutions to KdV and for the narrow wedge solution to KPZ
• Differential equations for the KPZ and periodic KPZ fixed points
• KPZ fluctuations in finite volume
• Jánossy densities and Darboux transformations for the Stark and cylindrical KdV equations
• Exact solution of interacting particle systems related to random matrices
• Large deviations for the $q$-deformed polynuclear growth