Speed and  fluctuations for some driven dimer models

Chhita, Sunil; Ferrari, Patrik L.; Toninelli, Fabio Lucio

doi:10.4171/aihpd/77

Cited by 9 publications

(13 citation statements)

References 25 publications

(73 reference statements)

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“…(ii) If instead det H ρ ≤ 0 ("Anisotropic KPZ" or AKPZ class) one expects that α = β = 0 and that moreover the growth of (1.2), (1.3) is logarithmic, exactly like for the two-dimensional stochastic heat equation. This belief is supported by the mathematical analysis of various (2 + 1)-dimensional growth models [2,5,7,33] that share the following features: stationary states can be found explicitly and their height fluctuations behave on large space scales as a massless Gaussian Field (GFF), with α = 0 and logarithmic correlations; the speed of growth v(·) can be computed and det H ρ turns out to be negative; the growth exponent β is zero and the height variance grows at most logarithmically with time. This state of affairs naturally leads to two questions.…”

Section: Introductionmentioning

confidence: 91%

“…We begin by recalling how to express probabilities of local events via the Kasteleyn matrix and how to rewrite its matrix elements as single integrals in the complex plane, as was done in [6]. We adopt a similar coordinate system to the one used in [6] but we have chosen to interchange the white and black vertices, so that we can keep the same height change conventions from a previous paper [5]. Namely, with reference to Fig.…”

Section: Proof Of Theorem 310mentioning

confidence: 99%

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

Chhita

Toninelli

2019

Commun. Math. Phys.

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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 ρ of the speed of growth v(ρ) as a function of the average slope ρ satisfies det H ρ > 0 ("isotropic KPZ class") or det H ρ ≤ 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 H ρ 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 translationinvariant Gibbs measures on perfect matchings of Z 2 , 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 H ρ < 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 H ρ for ρ ∼ 0. 1 arXiv:1802.05493v2 [math.PR]

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Section: Introductionmentioning

confidence: 91%

Section: Proof Of Theorem 310mentioning

confidence: 99%

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

Chhita

Toninelli

2019

Commun. Math. Phys.

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“…• in the statement of [43, Th. 3.1] there is a technical restriction on the slope ρ, that was later removed in joint work with S. Chhita and P. Ferrari [9]; • the proof that the speed of growth v(·) in (2.9) is the same as the function v(·) in (2.7), as it should, was obtained by S. Chhita and P. Ferrari in [8] and requires a nice combinatorial property of the Gibbs measures π ρ . With reference to Remark 2.3 above, it is important to emphasize that there is no known determinantal form for the space-time correlations of the stationary process; for the proof of (2.8) we used a more direct and probabilistic method.…”

Section: 2mentioning

confidence: 99%

(2 + 1)-Dimensional Interface Dynamics: Mixing Time, Hydrodynamic Limit and Anisotropic KPZ Growth

Toninelli

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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Stochastic interface dynamics serve as mathematical models for diverse time-dependent physical phenomena: the evolution of boundaries between thermodynamic phases, crystal growth, random deposition... 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. In contrast with the case of (1 + 1)-dimensional models, there are very few mathematical results in dimension (d+1), d ≥ 2. As far as growth models are concerned, the (2 + 1)-dimensional case is particularly interesting: Wolf [45] conjectured the existence of two different universality classes (called KPZ and Anisotropic KPZ), with different scaling exponents. 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. In particular, in the irreversible case, we discuss mathematical support and remaining open problems concerning Wolf's conjecture [45] on the relation between the Hessian of the growth velocity on one side, and the universality class of the model on the other.2010 Mathematics Subject Classification: 82C20, 60J10, 60K35, 82C24

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“…As we mentioned in the introduction, the two-dimensional domino-tiling growth model defined in Section 3.1 of [33] admits the Gibbs measures π ρ of the dimer model on Z 2 as stationary states. The speed of growth v(·) was computed in [12]: while it has quite a complicated expression in terms of ∇h, see [12, Eq. (2.6)], once expressed in terms of z it equals just π −1 z.…”

Section: A Couple Of Concrete Examplesmentioning

confidence: 99%

“…The first one is an AKPZ growth model based on domino tilings in the plane, defined in Section 3.1 of [33], that at first sight does not seem to fit into the general formalism of [7]. The proof of logarithmic growth fluctuations, implying α = β = 0, as well as the computation of the speed of growth and a direct verification that det(D 2 v(ρ)) < 0 can be found in [12]. The second one is the so-called q-Whittaker process, introduced in [4] in the wider context of Macdonald processes.…”

Section: Introductionmentioning

confidence: 99%

Two-dimensional anisotropic KPZ growth and limit shapes

Borodin¹,

Toninelli²

2018

J. Stat. Mech.

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A series of recent works focused on two-dimensional 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. In agreement with the scenario conjectured by D. Wolf [35], in all known AKPZ examples the function v(ρ) giving the growth velocity as a function of the slope ρ has a Hessian with negative determinant ("AKPZ signature"). 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 two-dimensional interface models. 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 v with respect to a natural complex structure.

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Speed and fluctuations for some driven dimer models

Cited by 9 publications

References 25 publications

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

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

(2 + 1)-Dimensional Interface Dynamics: Mixing Time, Hydrodynamic Limit and Anisotropic KPZ Growth

Two-dimensional anisotropic KPZ growth and limit shapes

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