Gaussian fluctuations from the 2D KPZ equation



We prove the two dimensional KPZ equation with a logarithmically tuned nonlinearity and a small coupling constant, scales to the Edwards–Wilkinson equation with an effective variance.


KPZ equation Edwards–Wilkinson equation Feynman–Kac formula 

Mathematics Subject Classification

35R60 60H07 60H15 



We would like to thank Li-Cheng Tsai for his initial involvement in this project and multiple inspiring discussions. We thank Nikolaos Zygouras for some helpful discussions, and two anonymous referees for a very careful reading of the manuscript and many helpful suggestions to improve the presentation. The research is supported by NSF Grant DMS-1613301/1807748/1907928 and the Center for Nonlinear Analysis of CMU.


  1. 1.
    Bertini, L., Cancrini, N.: The two-dimensional stochastic heat equation: renormalizing a multiplicative noise. J. Phys. A Math. Gen. 31, 615 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Borodin, A., Corwin, I., Ferrari, P.: Anisotropic \((2+1)d\) growth and Gaussian limits of \(q\)-Whittaker processes. Probab. Theory Relat. Fields 172, 245–321 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Borodin, A., Corwin, I., Toninelli, F.: Stochastic heat equation limit of a \((2+1)d\) growth model. Commun. Math. Phys. 350, 957–984 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Caravenna, F., Sun, R., Zygouras, N.: Universality in marginally relevant disordered systems. Ann. Appl. Probab. 27, 3050–3112 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Caravenna, F., Sun, R., Zygouras, N.: On the moments of the \((2+1)\)-dimensional directed polymer and stochastic heat equation in the critical window. arXiv:1808.03586 (2018)
  6. 6.
    Caravenna, F., Sun, R., Zygouras, N.: The two-dimensional KPZ equation in the entire subcritical regime. arXiv:1812.03911 (2018)
  7. 7.
    Chatterjee, S., Dunlap, A.: Constructing a solution of the \((2+ 1) \)-dimensional KPZ equation. arXiv preprint arXiv:1809.00803 (2018)
  8. 8.
    Chatterjee, S.: Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Relat. Fields 143, 1–40 (2009)CrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, Y.-T.: Rescaled Whittaker driven stochastic differential equations converge to the additive stochastic heat equation. Electron. J. Probab. 24(36), 1–33 (2019)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Comets, F., Cosco, C., Mukherjee, C.: Fluctuation and rate of convergence for the stochastic heat equation in weak disorder. arXiv preprint arXiv:1807.03902 (2018)
  11. 11.
    Comets, F., Cosco, C., Mukherjee, C.: Renormalizing the Kardar–Parisi–Zhang equation in \(d\ge 3\) in weak disorder. arXiv preprint arXiv:1902.04104 (2019)
  12. 12.
    Corwin, I.: The Kardar–Parisi–Zhang equation and universality class. Random Matrices Theory Appl. 1, 1130001 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Corwin, I., Shen, H.: Some recent progress in singular stochastic PDEs. arxiv preprint arXiv:1904.00334 (2019)
  14. 14.
    Cosco, C., Nakajima, S.: Gaussian fluctuations for the directed polymer partition function for \(d\ge 3\) and in the whole \(L^2\)-region. arXiv preprint arXiv:1903.00997 (2019)
  15. 15.
    Dunlap, A., Gu, Y., Ryzhik, L., Zeitouni, O.: Fluctuations of the solutions to the KPZ equation in dimensions three and higher. arXiv preprint arXiv:1812.05768 (2018)
  16. 16.
    Feng, Z.S.: Rescaled Directed Random Polymer in Random Environment in Dimension 1+2. Ph.D. thesis, University of Toronto, Canada (2016)Google Scholar
  17. 17.
    Gu, Y., Ryzhik, L., Zeitouni, O.: The Edwards–Wilkinson limit of the random heat equation in dimensions three and higher. Commun. Math. Phys. 363, 351–388 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hu, Y., Lê, K.: Asymptotics of the density of parabolic Anderson random fields. arXiv preprint arXiv:1801.03386 (2018)
  19. 19.
    Kallianpur, G., Robbins, H.: Ergodic property of the Brownian motion process. Proc. Natl. Acad. Sci. USA 39, 525–533 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Magnen, J., Unterberger, J.: The scaling limit of the KPZ equation in space dimension 3 and higher. J. Stat. Phys. 171, 543–598 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mukherjee, C., Shamov, A., Zeitouni, O.: Weak and strong disorder for the stochastic heat equation and continuous directed polymers in \( d\ge 3\). Electron. Commun. Probab. 21, 61 (2016)CrossRefzbMATHGoogle Scholar
  22. 22.
    Nourdin, I., Peccati, G., Reinert, G.: Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos. Ann. Probab. 38, 1947–1985 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Nourdin, I., Peccati, G., Reinert, G.: Second order Poincaré inequalities and CLTs on Wiener space. J. Funct. Anal. 257, 593–609 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Nualart, D., Peccati, G.: Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33, 177–193 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Quastel, J., Spohn, H.: The one-dimensional KPZ equation and its universality class. J. Stat. Phys. 160, 965–984 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Toninelli, F.: \((2+1)\)-dimensional interface dynamics: mixing time, hydrodynamic limit and Anisotropic Kardar–Parisi–Zhang growth. In: Proceedings of the International Congress of Mathematicians 2018, Rio de Janeiro, vol. 2, pp. 2719–2744. arXiv:1711.05571

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Authors and Affiliations

  1. 1.Department of MathematicsCarnegie Mellon UniversityPittsburghUSA

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