Long-range temporal correlations in the Kardar–Parisi–Zhang growth: numerical simulations

Song, Tianshu; Xia, Hui

doi:10.1088/1742-5468/2016/11/113206

Cited by 14 publications

(14 citation statements)

References 23 publications

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“…On the analytical side, the situation is thus unclear. Moreover, the very few existing numerical simulations [36,37] can not convincingly discriminate between the two scenarii (presence or absence of a threshold) nor on the sense of variation of z LR . They essentially find a very weak dependence at small θ and are too scattered to settle whether z LR is decreasing or increasing at larger values of θ.…”

Section: Introductionmentioning

confidence: 94%

“…In contrast, temporally correlated noise has received much less attention. The few existing analytical [15,[32][33][34][35] and numerical [36,37] studies yield conflicting results. One of the reasons is that the presence of temporal correlations is much more severe, in that it breaks the constitutive KPZ symmetry, which is the Galilean invariance, also known as statistical tilt symmetry.…”

Section: Introductionmentioning

confidence: 99%

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Kardar-Parisi-Zhang equation with temporally correlated noise: A nonperturbative renormalization group approach

Squizzato

Canet

2019

Phys. Rev. E

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We investigate the universal behavior of the Kardar-Parisi-Zhang equation with temporally correlated noise. The presence of time correlations in the microscopic noise breaks the statistical tilt symmetry, or Galilean invariance, of the original KPZ equation with delta-correlated noise (denoted SR-KPZ). Thus it is not clear whether the KPZ universality class is preserved in this case. Conflicting results exist in the literature, some advocating that it is destroyed even in the limit of infinitesimal temporal correlations, while others find that it persists up to a critical range of such correlations. Using non-perturbative and functional renormalization group techniques, we study the influence of two types of temporal correlators of the noise: a short range one with a typical time-scale τ , and a power-law one with a varying exponent θ. We show that for the short-range noise with any finite τ , the symmetries (the Galilean symmetry, and the time-reversal one in D = 1 + 1) are dynamically restored at large scales, and the long-distance properties are governed by the SR-KPZ fixed point. In the presence of a power-law noise, we find that the SR-KPZ fixed point is still stable for θ below a critical value θc, in accordance with previous RG results, while a long-range (LR) fixed-point controls the critical scaling for θ > θc, and we evaluate the θ-dependent critical exponents at this LR fixed point, in both D = 1 + 1 and D = 2 + 1 dimensions. While the results in D = 1 + 1 can be compared to previous estimates, no other prediction was available in D = 2 + 1.

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Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Kardar-Parisi-Zhang equation with temporally correlated noise: A nonperturbative renormalization group approach

Squizzato

Canet

2019

Phys. Rev. E

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“…Since the global roughness and dynamic exponents of the ALM are parametrized by n, however, one can ask whether a value of n exists such that the ALM and the quenched noise KPZ share the same exponents ζ, ζ s , and z. At this respect, numerical simulations of temporally correlated KPZ for d = 1 and λ > 0 report, for the φ = 1/2 quenched-noise limit, ζ ≈ 1.06, z ≈ 1.54 [36] and ζ ≈ 1.07, z ≈ 1.15 [37]. Exponents are thus roughly consistent with the n ≈ 4-5 ALM, where ζ ≈ 1.06 ± 0.01 and z ≈ 1.56.…”

Section: B Relation To Variants Of the Kpz Equationmentioning

confidence: 99%

Roughening of the anharmonic Larkin model

2019

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We study the roughening of d-dimensional directed elastic interfaces subject to quenched random forces. As in the Larkin model, random forces are considered constant in the displacement direction and uncorrelated in the perpendicular direction. The elastic energy density contains an harmonic part, proportional to (∂xu) 2 , and an anharmonic part, proportional to (∂xu) 2n , where u is the displacement field and n > 1 an integer. By heuristic scaling arguments, we obtain the global roughness exponent ζ, the dynamic exponent z, and the harmonic to anharmonic crossover length scale, for arbitrary d and n, yielding an upper critical dimension dc(n) = 4n. We find a precise agreement with numerical calculations in d = 1. For the d = 1 case we observe, however, an anomalous "faceted" scaling, with the spectral roughness exponent ζs satisfying ζs > ζ > 1 for any finite n > 1, hence invalidating the usual single-exponent scaling for two-point correlation functions, and the small gradient approximation of the elastic energy density in the thermodynamic limit. We show that such d = 1 case is directly related to a family of Brownian functionals parameterized by n, ranging from the random-acceleration model for n = 1, to the Lévy arcsine-law problem for n = ∞. Our results may be experimentally relevant for describing the roughening of nonlinear elastic interfaces in a Matheron-de Marsilly type of random flow.

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“…For DPRM, such a slow decay of correlations was proven to be relevant to the large-scale behavior, both in numerical simulations [23,25,34] and analytic calculations [21,24,35,36]. For Gaussian noise with isotropic correlations decaying as a power law with exponents between −0.5 and −0.2 (as measured for the road density correlations in Fig.…”

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confidence: 80%

Optimal paths on the road network as directed polymers

et al. 2017

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We analyze the statistics of the shortest and fastest paths on the road network between randomly sampled end points. We find that, to a good approximation, the optimal paths can be described as directed polymers in a disordered medium, which belong to the Kardar-Parisi-Zhang universality class of interface roughening. Comparing the scaling behavior of our data with simulations of directed polymers and previous theoretical results, we are able to point out the few characteristics of the road network that are relevant to the large-scale statistics of optimal paths. Indeed, we show that the local structure is akin to a disordered environment with a power-law distribution which become less important at large scales where long-ranged correlations in the network control the scaling behavior of the optimal paths. DOI: 10.1103/PhysRevE.96.050301 Complex networks of nodes and links can be used to model a wide array of systems. Examples range from biological networks such as those formed by neurons and synapses in the brain or chemical reactions inside a cell, to social or transportation networks and the World Wide Web. Their topology in the abstract space of edges and vertices has been much studied, allowing one to identify widespread properties such as "small-world" effects, scale-free connectivity, and a high degree of clustering, which can be captured by simple physical models [1][2][3][4][5]. Comparatively, less is understood about the spatial organization of complex networks embedded in a Euclidean space, a very active subject of research (see Ref.[6] for a review). The effect of geometry becomes especially relevant when the network is strongly constrained by the environment or when the "cost" to maintain edges increases significantly with their length (e.g., rivers [7], railways [8], or vascular networks [9]). The spatial structure of streets is another example that has been particularly studied to gain insight into the structure of cities and their development [10][11][12].Much information about the geometry of a network can be obtained by studying the shortest paths between the nodes of the network. In many cases, it is also a problem of practical importance to characterize the paths that optimize a given cost function. For example, in transportation networks, one would like to understand the properties of the paths that minimize the travel time, the distance, or the monetary cost to travel between two points. An obvious application is in the development of efficient global positioning system (GPS) routing algorithms which could use prior information on optimal paths to perform better [13]. The shortest paths between two generating nodes on the power grid are also important to predict the overloading of electric lines [14]. Understanding the properties of these optimal paths appears challenging since they are expected to depend strongly on the geometry of the network which can be shaped by various factors, from natural obstacles to historical development or differences in policy.The theory of directed polymers...

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Long-range temporal correlations in the Kardar–Parisi–Zhang growth: numerical simulations

Cited by 14 publications

References 23 publications

Kardar-Parisi-Zhang equation with temporally correlated noise: A nonperturbative renormalization group approach

Kardar-Parisi-Zhang equation with temporally correlated noise: A nonperturbative renormalization group approach

Roughening of the anharmonic Larkin model

Optimal paths on the road network as directed polymers

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