A System of PDEs for the Baik-Rains Distribution

TL;DR

This paper proves that the anti-derivative of the Baik-Rains distribution, relevant for KPZ universality class fluctuations, satisfies the KP equation, linking stochastic growth models to integrable PDEs.

Contribution

It establishes that the Baik-Rains distribution's anti-derivative obeys the KP equation, connecting stationary initial data fluctuations to integrable PDEs.

Findings

Baik-Rains distribution's anti-derivative satisfies the KP equation.

The KP equation governs the distribution of stationary KPZ fluctuations.

Large time limit preserves the KP equation for the generating function.

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 Baik-Rains distribution, which governs the fluctuation of the models in the KPZ universality class starting with stationary initial data, satisfies the KP equation. We start from a determinantal formula of the generating function of the KPZ equation, which satisfies the KP. Then we observe that the equation still holds in the large time limit.