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The Scaling Limit of the \((\nabla +\Delta )\)-Model

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Abstract

In this article we study the scaling limit of the interface model on \({{\,\mathrm{{\mathbb {Z}}}\,}}^d\) where the Hamiltonian is given by a mixed gradient and Laplacian interaction. We show that in any dimension the scaling limit is given by the Gaussian free field. We discuss the appropriate spaces in which the convergence takes place. While in infinite volume the proof is based on Fourier analytic methods, in finite volume we rely on some discrete PDE techniques involving finite-difference approximation of elliptic boundary value problems.

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Authors and Affiliations

  1. TU Delft (DIAM), Building 28, van Mourik Broekmanweg 6, 2628, XE, Delft, The Netherlands

    Alessandra Cipriani

  2. ISI Kolkata, 203, B.T. Road, Kolkata, 700108, India

    Biltu Dan & Rajat Subhra Hazra

Authors
  1. Alessandra Cipriani
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  2. Biltu Dan
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  3. Rajat Subhra Hazra
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Correspondence to Alessandra Cipriani.

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Communicated by Eric A. Carlen.

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AC is supported by grant 613.009.102 of the Netherlands Organisation for Scientific Research (NWO). RSH acknowledges MATRICS grant from SERB and the Dutch stochastics cluster STAR (Stochastics – Theoretical and Applied Research) for an invitation to TU Delft where part of this work was carried out. The authors thank Francesco Caravenna for helpful discussions, and an anonymous referee for insightful remarks and comments on a previous draft of the work.

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Cipriani, A., Dan, B. & Hazra, R.S. The Scaling Limit of the \((\nabla +\Delta )\)-Model. J Stat Phys 182, 39 (2021). https://doi.org/10.1007/s10955-021-02717-1

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