Surface roughening in a hypercube-stacking model

Forrest, Bruce M.; Tang, Lei-Han

doi:10.1103/physrevlett.64.1405

Cited by 203 publications

(141 citation statements)

References 26 publications

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“…9 in that paper. Interestingly, the 2+1 KPZ stationary-state has also permitted a solid multi-model (gDPRM,RSOS,KPZ Euler) estimate β 2+1 =0.241(1), in line with prior gold-medal studies [20,124] at the 3-digit precision level.…”

Section: Universal Limit Distribution: 3d Radial Kpz Classmentioning

confidence: 76%

“…• Recalling recent 2+1 DPRM results [84], which have isolated the higher-dimensional analog of KPZ/TW-GOE, measuring s=0.424 and k=0.346 for the generic pt-plane case, and pinned down the key index ω 2+1 =0.241 [20,124] via a multi-model study [85] of 2+1 KPZ stationary-state statistics, we notice quite similar values for the b=2.3 ♦DPRM, where the exponent, 0.274, might be a little high, but the skewness, 0.423, and kurtosis, 0.372, certainly close to the mark. Given our findings above for the ♦DPRM sk relation, one cannot resist looking at Euclidean KPZ from this same vantage point.…”

Section: The Many-dimensional Dprm and Fate Of D=∞ Kpzmentioning

confidence: 99%

“…Here, ν, λ and D are coarse-grained, macroscopic phenomenological parameters, calculable for any given microscopic model via the technical machinery of KPZ scaling theory, elucidated by Krug, Meakin and HalpinHealy (KMHH) [2,3], built upon earlier finite-size scaling ideas [4], but predicated ultimately on the central role of the key scaling parameter θ = A 1/α λ, where A = D/2ν, and a divergent, parallel correlation length ξ = ( √ Aλt) 1/z . The 1 st KPZ Revolution, 1985-1999, was characterized by exact [5,6,7,8,9,10,11,12], renormalization group [13,14], mode-coupling [15,16], and numerical investigations [17,18,19,20,21,22,23], whose primary focus was pinning down the dimension-dependent saturationwidth and transient-regime scaling indices: α and β= α z . Galilean invariance, a key underlying symmetry of the KPZ equation was guarantor of the sacred, dimension-independent index identity α + z = 2.…”

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A KPZ Cocktail-Shaken, not Stirred...

2015

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The stochastic partial differential equation proposed nearly three decades ago by Kardar, Parisi and Zhang (KPZ) continues to inspire, intrigue and confound its many admirers. Here, we i) pay debts to heroic predecessors, ii) highlight additional, experimentally relevant aspects of the recently solved 1+1 KPZ problem, iii) use an expanding substrates formalism to gain access to the 3d radial KPZ equation and, lastly, iv) examining extremal paths on disordered hierarchical lattices, set our gaze upon the fate of d=∞ KPZ. Clearly, there remains ample unexplored territory within the realm of KPZ and, for the hearty, much work to be done, especially in higher dimensions, where numerical and renormalization group methods are providing a deeper understanding of this iconic equation.Keywords Nonequilibrium Growth · Extremal Paths · Universal Limit Distributions In a NutshellThe history of physics has been punctuated at seminal moments by the appearance of certain fundamental equations (and associated models) which have vigorously propelled the enterprise forward, serving as an explosively rich departure point, generating a myriad of alternative perspectives, creative insights, surprising connections and, given its sustained impregnability, often remain for many years a sacred object of fascination and obsession to its dedicated disciples. Recent, obvious suspects in this regard include the quantum mechanical Schrödinger equation (equally well, its flipside-Feynman's path integral formulation), or the wonderfully elusive Navier-Stokes equation governing fluid mechanics, by which may be gleaned the scaling secrets of turbulent flow, dynamically encoded in the whirling eddies drawn by da Vinci centuries ago. Within the domain of equilibrium statistical physics, the 2d Ising Model, with its tour-de-force algebraic solution by Onsager, followed by combinatoric, graphical, Grassmannian, Monte Carlo, as well as full-blown field-theoretic, scaling, and renormalization group treatments, represents an extraordinary legacy that continues unabated to this very day. Arguably, a non-equilibrium statistical mechanical analogue to Ising/Onsager is the iconic equation [1] proposed a generation ago by Kardar, Parisi, and Zhang (KPZ); it captures the statistical fluctuations of a kinetically-roughened scalar height field h(x, t):

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Section: Universal Limit Distribution: 3d Radial Kpz Classmentioning

confidence: 76%

Section: The Many-dimensional Dprm and Fate Of D=∞ Kpzmentioning

confidence: 99%

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confidence: 99%

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A KPZ Cocktail-Shaken, not Stirred...

2015

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“…In the two dimensional case, the values of the exponents are ν ≃ 2/3 , ω ≃ 1/3, and the values 2/3 and 1/3 are considered to be exact. In dimensions higher then 2, there are only numerical estimates for the values of the exponents, and in the three and four dimensional cases, the accepted values for the exponents are ν ≃ 0.62, ω ≃ 0.24, and ν ≃ 0.59, ω ≃ 0.18 respectively [10,11]. However, these numerical estimates were obtained for random values taken from Gaussian distribution, while non-Gaussian distributions, though preserving the space exponents [11], yield lower estimates for the energy exponents [5,11].…”

Section: 40-amentioning

confidence: 99%

Universality of the directed polymer model

Perlsman

Havlin

2000

Phys. Rev. E

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The universality of the directed polymer model and the analogous KPZ equation is supported by numerical simulations using non-Gaussian random probability distributions in two, three and four dimensions. It is shown that although in the non-Gaussian cases the finite size estimates of the energy exponents are below the persumed universal values, these estimates increase with the system size, and the further they are below the universal values, the higher is their rate of increase. The results are explained in terms of the efficiency of variance reduction during the optimization process. 05.40-aThe directed polymer model [1] and the analogous KPZ equation [2] have drawn significant attention recently [3]. The original introduction of the model [1] considered directed polymers in random media, and more recently it has been applied to model other processes such as tearing and cracks formation [4]. The study of the model is focused on the characteristics of the optimal path which connects two distant points in a random environment. Passing through any point of that environment is associated with a random valued cost, usually expressed in units of energy or time. Unlike the simple case of constant cost, the path of minimal cost (the optimal path) is usually not the straight line which connects the two points, and its mean transversal distance from the straight line which connects its endpoints grows as a power law with the distance between the endpoints. Another power law behaviour is associated with the relation between the variance of the energy cost (or time cost), and the distance between the endpoints. These two power laws are expressed in terms of their exponents, which are connected by a scaling relation. The random environment is usually simulated by a lattice whose bonds (or sites) are assigned with random values, and it is generally believed that the characteristics of the optimal path are universal, i.e. apart from very special cases, the exponents depend only on the dimension of the lattice and not on the details of the lattice structure, or the type of randomness associated with it.The universality hypothesis was recently challenged by numerical results [5] which showed that for dimensions higher than 2, the values of the estimated exponents do depend on the type of the probability distribution which is associated with the random lattice. In particular, nonGaussian distributions yield lower values for the exponents compared to Gaussian distribution values, which are the persumed universal values. This publication has raised strong opposition in the form of three recent articles [6,7,8], which argued that the estimates presented in [5] are affected by finite size effects, and thus they are not the asymptotic values of the exponents in these cases. Two of these articles [6,7] independently suggested that the estimates presented in [5] are disturbed by directed percolation effects. However, none of these opposing articles presented any alternative numerical estimates to those presented in [5].This article pre...

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“…In a recent Letter, Newman and Swift [1] made an interesting suggestion that the strong-coupling exponents of the Kardar-Parisi-Zhang [2] (KPZ) equation reported previously [3,4] may not be universal, but rather depend on the precise form of the noise distribution. The purpose of this Comment is to show that their numerical findings can be attributed to a percolative effect instead of a portentous breakdown of universality.…”

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confidence: 99%