Open Access
2022 The logarithmic anti-derivative of the Baik-Rains distribution satisfies the KP equation
Xincheng Zhang
Author Affiliations +
Electron. Commun. Probab. 27: 1-12 (2022). DOI: 10.1214/22-ECP469


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.


I am very grateful to my supervisor Professor Jeremy Quastel for suggesting this problem to me. He gave me many invaluable guidance and discussions on this topic, besides he gave me many important suggestions on my writing of the paper.


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Xincheng Zhang. "The logarithmic anti-derivative of the Baik-Rains distribution satisfies the KP equation." Electron. Commun. Probab. 27 1 - 12, 2022.


Received: 8 March 2021; Accepted: 9 May 2022; Published: 2022
First available in Project Euclid: 18 May 2022

MathSciNet: MR4424036
zbMATH: 1496.60118
Digital Object Identifier: 10.1214/22-ECP469

Primary: 60

Keywords: Limiting distribution , the KPZ equation

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