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KPZ equation with a small noise, deep upper tail and limit shape

Abstract

In this paper, we consider the KPZ equation under the weak noise scaling. That is, we introduce a small parameter \(\sqrt{\varepsilon }\) in front of the noise and let \(\varepsilon \rightarrow 0\). We prove that the one-point large deviation rate function has a \(\frac{3}{2}\) power law in the deep upper tail. Furthermore, by forcing the value of the KPZ equation at a point to be very large, we prove a limit shape of the solution of the KPZ equation as \(\varepsilon \rightarrow 0\). This confirms the physics prediction in Hartmann et al. (Phys Rev Res 1(3):032043, 2019), Kolokolov and Korshunov (Phys Rev B 75(14):140201, 2007, Phys Rev E 80(3):031107, 2009), Kamenev et al. (Phys Rev E 94(3):032108, 2016), Le Doussal et al. (Phys Rev Lett 117(7):070403, 2016) and Meerson et al. (Phys Rev Lett 116(7):070601, 2016).

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Acknowledgements

We thank Ivan Corwin, Alexandre Krajenbrink, Pierre Le Doussal, and Baruch Meerson for their helpful comments on the presentation of this work. We thank the referees for their useful comments on the manuscript, especially for pointing out an error in Lemma 3.1 in the first version of the manuscript. The research of LCT is partially supported by the Sloan Fellowship and the NSF through DMS-1953407 and DMS-2153739.

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Correspondence to Yier Lin.

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Gaudreau Lamarre, P.Y., Lin, Y. & Tsai, LC. KPZ equation with a small noise, deep upper tail and limit shape. Probab. Theory Relat. Fields 185, 885–920 (2023). https://doi.org/10.1007/s00440-022-01185-2

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  • DOI: https://doi.org/10.1007/s00440-022-01185-2

Mathematics Subject Classification

  • 60F10
  • 60H15