![]() ![]() Here, we review recent mathematical results on (both reversible and irreversible) dynamics of some $(2 1)$-dimensional discrete interfaces, mostly defined through a mapping to two-dimensional dimer models. Wolf conjectured the existence of two different universality classes (called KPZ and Anisotropic KPZ), with different scaling exponents. ![]() As far as growth models are concerned, the $(2 1)$-dimensional case is particularly interesting: D. In contrast with the case of $(1 1)$-dimensional models, there are very few mathematical results in dimension $(d 1), d\ge2$. Interesting limits arise at large space-time scales: after suitable rescaling, the randomly evolving interface converges to the solution of a deterministic PDE (hydrodynamic limit) and the fluctuation process to a (in general non-Gaussian) limit process. ![]() Stochastic interface dynamics serve as mathematical models for diverse time-dependent physical phenomena: the evolution of boundaries between thermodynamic phases, crystal growth, random deposition. In the case of growth processes defined via dynamics of dimer models on planar lattices, we further prove that the preservation of the Euler-Lagrange equations is equivalent to harmonicity of with respect to a natural complex structure. While up to now negativity was verified model by model via explicit computations, in this work we show that it actually has a simple geometric origin in the fact that the hydrodynamic PDEs associated to these non-equilibrium growth models preserves the Euler-Lagrange equations determining the macroscopic shapes of certain equilibrium 2D interface models. 67 1783-6), in all known AKPZ examples the function giving the growth velocity as a function of the slope ρ has a Hessian with negative determinant ('AKPZ signature'). In agreement with the scenario conjectured by Wolf (1991 Phys. #Circuli ambient music creator series#In contrast with the hydrodynamic equation for the Langevin dynamics of the Ginzburg-Landau model, here the mobility coefficient turns out to be a non-trivial function of the interface slope.Ī series of recent works focused on two-dimensional (2D) interface growth models in the so-called anisotropic KPZ (AKPZ) universality class, that have a large-scale behavior similar to that of the Edwards-Wilkinson equation. The explicit form of the PDE was recently conjectured on the basis of local equilibrium considerations. ![]() In this work we prove a hydrodynamic limit: after a diffusive rescaling of time and space, the height function evolution tends as $L\to\infty$ to the solution of a non-linear parabolic PDE. This dynamics presents special features: the average mutual volume between two interface configurations decreases with time and a certain one-dimensional projection of the dynamics is described by the heat equation. We consider a particular choice of the transition rates, originally proposed in : in terms of interlaced particles, a particle jump of length $n$ that preserves the interlacement constraints has rate $1/(2n)$. The particle interlacement constraints imply that the equilibrium measures are far from being product Bernoulli: particle correlations decay like the inverse distance squared and interface height fluctuations behave on large scales like a massless Gaussian field. This can be alternatively viewed as a dynamics of lozenge tilings of the $L\times L$ torus, or as a conservative dynamics for a two-dimensional system of interlaced particles. We study a reversible continuous-time Markov dynamics of a discrete $(2 1)$-dimensional interface. ![]()
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