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A (2 + 1)-dimensional anisotropic KPZ growth model with a smooth phase

Chhita, Sunil; Toninelli, Fabio Lucio

A (2 + 1)-dimensional anisotropic KPZ growth model with a smooth phase Thumbnail


Fabio Lucio Toninelli


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 Hρ of the speed of growth v(ρ) as a function of the average slope ρ satisfies det Hρ > 0 (“isotropic KPZ class”) or det Hρ ≤ 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(·) is not differentiable, so that Hρ 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 Z2, with 2-periodic weights. If ρ = 0, 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 det Hρ < 0: the model belongs to the AKPZ class. When ρ = 0, instead, the stationary state is “smooth”, with correlations uniformly bounded in space and time; correspondingly, v(·) is not differentiable at ρ = 0 and we extract the singularity of the eigenvalues of Hρ for ρ ∼ 0.


Chhita, S., & Toninelli, F. L. (2019). A (2 + 1)-dimensional anisotropic KPZ growth model with a smooth phase. Communications in Mathematical Physics, 367(2), 483-516.

Journal Article Type Article
Acceptance Date Oct 10, 2018
Online Publication Date Mar 26, 2019
Publication Date Apr 30, 2019
Deposit Date Sep 24, 2018
Publicly Available Date Apr 17, 2019
Journal Communications in Mathematical Physics
Print ISSN 0010-3616
Electronic ISSN 1432-0916
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 367
Issue 2
Pages 483-516
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Copyright Statement
© The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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