Communications in Mathematical Physics

, Volume 367, Issue 2, pp 483–516 | Cite as

A (2 + 1)-Dimensional Anisotropic KPZ Growth Model with a Smooth Phase

  • Sunil ChhitaEmail author
  • Fabio Lucio Toninelli
Open Access


Stochastic growth processes in dimension (2 + 1) were conjectured by D. Wolf, on the basis of renormalization-group arguments, to fall into two distinct universality classes, according to whether the Hessian \({H_\rho}\) of the speed of growth \({v(\rho)}\) as a function of the average slope \({\rho}\) satisfies \({{\rm det} H_\rho > 0}\) (“isotropic KPZ class”) or \({{\rm det} H_\rho \le 0}\) (“anisotropic KPZ (AKPZ)” class). The former is characterized by strictly positive growth and roughness exponents, while in the AKPZ class fluctuations are logarithmic in time and space. It is natural to ask (a) if one can exhibit interesting growth models with “smooth” stationary states, i.e., with O(1) fluctuations (instead of logarithmically or power-like growing, as in Wolf’s picture) and (b) what new phenomena arise when \({v(\cdot)}\) is not differentiable, so that \({H_\rho}\) is not defined. The two questions are actually related and here we provide an answer to both, in a specific framework. We define a (2 + 1)-dimensional interface growth process, based on the so-called shuffling algorithm for domino tilings. The stationary, non-reversible measures are translation-invariant Gibbs measures on perfect matchings of \({\mathbb{Z}^2}\) , with 2-periodic weights. If \({\rho\ne0}\) , fluctuations are known to grow logarithmically in space and to behave like a two-dimensional GFF. We prove that fluctuations grow at most logarithmically in time and that \({{\rm det} H_\rho < 0}\) : the model belongs to the AKPZ class. When \({\rho=0}\) , instead, the stationary state is “smooth”, with correlations uniformly bounded in space and time; correspondingly, \({v(\cdot)}\) is not differentiable at \({\rho=0}\) and we extract the singularity of the eigenvalues of \({H_\rho}\) for \({\rho\sim 0}\) .



We are grateful to Patrik Ferrari and Sanjay Ramassamy for enlightening discussions. We would also like to thank the referee for their careful reading of ourmanuscript. F.T. was partially supported by the CNRS PICS grant “Interfaces aléatoires discrètes et dynamiques de Glauber”, by the ANR-15-CE40-0020-03 Grant LSD and by Labex MiLyon (ANR-10-LABX-0070).


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Authors and Affiliations

  1. 1.Department of Mathematical SciencesDurham UniversityDurhamUK
  2. 2.UMR 5208, CNRS, Institut Camille Jordan, Univ LyonUniversité Claude Bernard Lyon 1Villeurbanne CedexFrance

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