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


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Article Info

DATE

2019-04

AUTHORS

Sunil Chhita, Fabio Lucio Toninelli

ABSTRACT

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 detHρ>0 (“isotropic KPZ class”) or detHρ≤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 detHρ<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 . More... »

PAGES

1-34

References to SciGraph publications

  • 1982-10. Surface tension, percolation, and roughening in JOURNAL OF STATISTICAL PHYSICS
  • 2014-01. Anisotropic Growth of Random Surfaces in 2 + 1 Dimensions in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1992-09. Alternating-Sign Matrices and Domino Tilings (Part I) in JOURNAL OF ALGEBRAIC COMBINATORICS
  • 1999. Scaling Limits of Interacting Particle Systems in NONE
  • 2018-10. The Edwards–Wilkinson Limit of the Random Heat Equation in Dimensions Three and Higher in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 2015-08. The One-Dimensional KPZ Equation and Its Universality Class in JOURNAL OF STATISTICAL PHYSICS
  • 1991. Large Scale Dynamics of Interacting Particles in NONE
  • 2015-09. Lozenge Tilings, Glauber Dynamics and Macroscopic Shape in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1982-03. Decay of correlations in surface models in JOURNAL OF STATISTICAL PHYSICS
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