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Stochastic Heat Equation Limit of a (2 + 1)d Growth Model

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Abstract

We determine a \({q \to 1}\) limit of the two-dimensional q-Whittaker driven particle system on the torus studied previously in Corwin and Toninelli (Electron. Commun. Probab. 21(44):1–12, 2016). This has an interpretation as a (2 + 1)-dimensional stochastic interface growth model, which is believed to belong to the so-called anisotropic Kardar–Parisi–Zhang (KPZ) class. This limit falls into a general class of two-dimensional systems of driven linear SDEs which have stationary measures on gradients. Taking the number of particles to infinity we demonstrate Gaussian free field type fluctuations for the stationary measure. Considering the temporal evolution of the stationary measure, we determine that along characteristics, correlations are asymptotically given by those of the (2 + 1)-dimensional additive stochastic heat equation. This confirms (for this model) the prediction that the non-linearity for the anisotropic KPZ equation in (2 + 1)-dimension is irrelevant.

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA

    Alexei Borodin

  2. Institute for Information Transmission Problems, Bolshoy Karetny per. 19, Moscow, 127994, Russia

    Alexei Borodin

  3. Department of Mathematics, Columbia University, 2990 Broadway, New York, NY, 10027, USA

    Ivan Corwin

  4. Clay Mathematics Institute, 10 Memorial Blvd. Suite 902, Providence, RI, 02903, USA

    Ivan Corwin

  5. Université de Lyon, CNRS and Institut Camille Jordan, Université Lyon 1, 43 bd du 11 novembre 1918, 69622, Villeurbanne, France

    Fabio Lucio Toninelli

Authors
  1. Alexei Borodin
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  2. Ivan Corwin
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  3. Fabio Lucio Toninelli
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Corresponding author

Correspondence to Ivan Corwin.

Additional information

Communicated by H. Spohn

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Borodin, A., Corwin, I. & Toninelli, F.L. Stochastic Heat Equation Limit of a (2 + 1)d Growth Model. Commun. Math. Phys. 350, 957–984 (2017). https://doi.org/10.1007/s00220-016-2718-4

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  • DOI: https://doi.org/10.1007/s00220-016-2718-4

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