Schramm-Loewner Evolution and isoheight lines of correlated landscapes

Posé, Nicolas; Schrenk, K. J.; Araújo, Nuno A. M.; Herrmann, Hans J.

doi:10.48550/arxiv.1508.07942

Cited by 2 publications

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“…Moreover, the SLE κ approach allows us to generate directly such conformally invariant curves without the need to generating correlated surfaces or simulate growth models. Taking advantage of this feature, several numerical and empirical studies of correlated random systems such as turbulent vorticity fields 7,8 , graphene sheets 9 , topology landscapes 10 , percolation in correlated surfaces 13 and accessible perimeters at fixed scale 14 , have analyzed the corresponding two-dimensional random curves in the context of…”

Section: Introductionmentioning

confidence: 99%

Schramm-Loewner evolution and perimeter of percolation clusters of correlated random landscapes

Castro

Luković

Pompanin

et al. 2018

Sci Rep

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Motivated by the fact that many physical landscapes are characterized by long-range height-height correlations that are quantified by the Hurst exponent H, we investigate the statistical properties of the iso-height lines of correlated surfaces in the framework of Schramm-Loewner evolution (SLE). We show numerically that in the continuum limit the external perimeter of a percolating cluster of correlated surfaces with H ∈ [−1, 0] is statistically equivalent to SLE curves. Our results suggest that the external perimeter also retains the Markovian properties, confirmed by the absence of time correlations in the driving function and the fact that the latter is Gaussian distributed for any specific time. We also confirm that for all H the variance of the winding angle grows logarithmically with size.

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Section: Introductionmentioning

confidence: 99%

Schramm-Loewner evolution and perimeter of percolation clusters of correlated random landscapes

Castro

Luković

Pompanin

et al. 2018

Sci Rep

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“…Random Gaussian surfaces (RGS) with positive Hurst exponents are examples of rough self-affine surfaces, and their use has become very popular since they are analytically tractable. Recently, it was suggested that iso-height lines in this type of RGS are not conformally invariant, 3 since their statistics is not compatible with the Schramm-Loewner Evolution (SLE) theory 4,[4][5][6][7][8] (random curves satisfying SLE statistics are necessarily conformally invariant). However, the fact that iso-height lines of other self-affine rough surfaces, e.g.…”

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confidence: 99%

Conformal Invariance of Graphene Sheets

Giordanelli

Posé

Mendoza

et al. 2016

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Suspended graphene sheets exhibit correlated random deformations that can be studied under the framework of rough surfaces with a Hurst (roughness) exponent 0.72 ± 0.01. Here, we show that, independent of the temperature, the iso-height lines at the percolation threshold have a well-defined fractal dimension and are conformally invariant, sharing the same statistical properties as SchrammLoewner evolution (SLE κ ) curves with κ = 2.24 ± 0.07. Interestingly, iso-height lines of other rough surfaces are not necessarily conformally invariant even if they have the same Hurst exponent, e.g. random Gaussian surfaces. We have found that the distribution of the modulus of the Fourier coefficients plays an important role on this property. Our results not only introduce a new universality class and place the study of suspended graphene membranes within the theory of critical phenomena, but also provide hints on the long-standing question about the origin of conformal invariance in isoheight lines of rough surfaces.Rough surfaces are very common in nature and can be found, for instance, in real landscapes 1 and growth surface processes 2 . In many cases, they can be characterised by a Hurst (roughness) exponent that describes the height-height correlations of the surface, and consequently, being called self-affine. Random Gaussian surfaces (RGS) with positive Hurst exponents are examples of rough self-affine surfaces, and their use has become very popular since they are analytically tractable. Recently, it was suggested that iso-height lines in this type of RGS are not conformally invariant 3 , since their statistics is not compatible with the Schramm-Loewner Evolution (SLE) theory 4,4-8 (random curves satisfying SLE statistics are necessarily conformally invariant). However, the fact that iso-height lines of other self-affine rough surfaces, e.g. some grown surfaces 9,10 , follow SLE theory opens the question on which are the main properties that are responsible for conformal invariance.Graphene, consisting of literally a single carbon monolayer, represents the first instance of a truly two-dimensional material (see Fig. 1) [11][12][13] . It owes its stability to the anharmonic coupling between bending and stretching modes, and the resulting deformation in the third dimension 14 . The study of these deformations, often called ripples, is very important because they affect the electronic and mechanical properties. For instance, it has been shown that reducing height fluctuations in graphene samples increases their electronic mobility and electrical conductivity 15 . Since previous numerical studies have shown that the structure of graphene possesses self-affine properties 16 , in this report, we show that iso-height lines, extracted at the percolation threshold, have a well-defined fractal dimension and are conformally invariant (note that to obtain the percolation threshold one starts from the highest point of the membrane and systematically lowers down the height until a cluster, formed by carbon atoms bounded by covalen...

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Schramm-Loewner Evolution and isoheight lines of correlated landscapes

Cited by 2 publications

References 1 publication

Schramm-Loewner evolution and perimeter of percolation clusters of correlated random landscapes

Schramm-Loewner evolution and perimeter of percolation clusters of correlated random landscapes

Conformal Invariance of Graphene Sheets

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