The logarithmic anti-derivative of the Baik-Rains distribution satisfies the KP equation

Abstract

It has been discovered that the Kadomtsev-Petviashvili (KP) equation governs the distribution of the fluctuation of many random growth models. In particular, the Tracy-Widom distributions appear as special self-similar solutions of the KP equation. We prove that the anti-derivative of the Baik-Rains distribution, which governs the fluctuation of the models in the KPZ universality class starting with stationary initial data, satisfies the KP equation. The result is first derived formally by taking a limit of the generating function of the KPZ equation, which satisfies the KP equation. Then we prove it directly using the explicit Painlevé II formulation of the Baik-Rains distribution. the long-time fluctuations of models which belong to the KPZ universality class. A 1 ( x ) is a stationary process, whose one-point distribution is the Tracy-Widom GOE distribution. The one point marginal of A 2 ( x ) is given by the Tracy-Widom GUE distribution. The one point marginal of A stat ( x ) is given by the Baik-Rains distribution.