Abstract:Many examples of natural systems can be described by random Gaussian surfaces. Much can be learned by analyzing the Fourier expansion of the surfaces, from which it is possible to determine the corresponding Hurst exponent and consequently establish the presence of scale invariance. We show that this symmetry is not affected by the distribution of the modulus of the Fourier coefficients. Furthermore, we investigate the role of the Fourier phases of random surfaces. In particular, we show how the surface is aff… Show more
“…For H = −1 and H = 0 we recover the analytical results that predict κ = 6 and κ = 4, respectively. Using the relationship between κ and d f demonstrated by Beffara 7 , we also show that the conjectured H -dependencies of the diffusivity κ and fractal dimension 4 , 8 , d f mutually corroborate each other. Finally, we also verify the Markov property of the curves by showing that the corresponding driving functions are uncorrelated in time and that they follow Gaussian statistics.…”
Section: Introductionsupporting
confidence: 72%
“…2 ) that the H -dependence of the complete perimeter fractal dimension has the form d f ( H ) = 3/2 − H /3 for H ∈ [−3/4, 0]. Moreover, the H -dependence was later also shown to be independent of the shape of the distribution of the random numbers, u ( q ), used to generate the correlated landscapes 8 . Figure 1 Examples of complete perimeters of percolating clusters for H = −0.1 and H = −1 and their respective driving functions, calculated by the zipper algorithm.…”
Section: Resultsmentioning
confidence: 99%
“…More precisely, we investigated whether the complete perimeter of a percolating cluster obeys the SLE κ statistics properties in the continuum limit. By taking the percolation threshold iso-height lines from correlated surfaces with −1 ≤ H ≤ 0 8 , we find that the lines indeed do follow SLE statistics. For H = −1 and H = 0 we recover the analytical results that predict κ = 6 and κ = 4, respectively.…”
Section: Introductionmentioning
confidence: 87%
“…As stated by the extended Harris criterion, if H ≤ −1/ ν uncorr then the correlations do not affect the critical exponents of the percolation transition. For 2D systems ν uncorr = 4/3 so that for H ∈ [−1, −3/4] the exponents are expected to be the same as for the uncorrelated system 4 , 8 , whereas for H ∈ [−3/4, 0] the critical exponents are expected to depend on H .…”
Section: Methodsmentioning
confidence: 99%
“…More precisely, the complete perimeter consists of all the lattice site edges that separate sites belonging to the percolating cluster from unoccupied sites that can be reached from the boundaries of the system without crossing the percolating cluster itself 4 . In the following study and analysis of the winding angle, direct SLE method and correlation time of the driving functions, we consider only the complete perimeter of the percolating cluster extracted from such correlated surfaces 8 .…”
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.
“…For H = −1 and H = 0 we recover the analytical results that predict κ = 6 and κ = 4, respectively. Using the relationship between κ and d f demonstrated by Beffara 7 , we also show that the conjectured H -dependencies of the diffusivity κ and fractal dimension 4 , 8 , d f mutually corroborate each other. Finally, we also verify the Markov property of the curves by showing that the corresponding driving functions are uncorrelated in time and that they follow Gaussian statistics.…”
Section: Introductionsupporting
confidence: 72%
“…2 ) that the H -dependence of the complete perimeter fractal dimension has the form d f ( H ) = 3/2 − H /3 for H ∈ [−3/4, 0]. Moreover, the H -dependence was later also shown to be independent of the shape of the distribution of the random numbers, u ( q ), used to generate the correlated landscapes 8 . Figure 1 Examples of complete perimeters of percolating clusters for H = −0.1 and H = −1 and their respective driving functions, calculated by the zipper algorithm.…”
Section: Resultsmentioning
confidence: 99%
“…More precisely, we investigated whether the complete perimeter of a percolating cluster obeys the SLE κ statistics properties in the continuum limit. By taking the percolation threshold iso-height lines from correlated surfaces with −1 ≤ H ≤ 0 8 , we find that the lines indeed do follow SLE statistics. For H = −1 and H = 0 we recover the analytical results that predict κ = 6 and κ = 4, respectively.…”
Section: Introductionmentioning
confidence: 87%
“…As stated by the extended Harris criterion, if H ≤ −1/ ν uncorr then the correlations do not affect the critical exponents of the percolation transition. For 2D systems ν uncorr = 4/3 so that for H ∈ [−1, −3/4] the exponents are expected to be the same as for the uncorrelated system 4 , 8 , whereas for H ∈ [−3/4, 0] the critical exponents are expected to depend on H .…”
Section: Methodsmentioning
confidence: 99%
“…More precisely, the complete perimeter consists of all the lattice site edges that separate sites belonging to the percolating cluster from unoccupied sites that can be reached from the boundaries of the system without crossing the percolating cluster itself 4 . In the following study and analysis of the winding angle, direct SLE method and correlation time of the driving functions, we consider only the complete perimeter of the percolating cluster extracted from such correlated surfaces 8 .…”
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.
Background: Digital Image Correlation (DIC) is based on the matching, between reference and deformed state images, of features contained in patterns that are deposited on test sample surfaces. These features are often suitable for a single scale, and there is a current lack of multiscale patterns capable of providing reliable displacement measurements over a wide range of scales. Objective: Here, we aim to demonstrate that a pattern based on a fractal (self-affine) surface would make a suitable pattern for multiscale DIC. Methods: A method to numerically generate patterns directly from a desired auto-correlation function is introduced. It is then enhanced by a Mean Intensity Gradient (MIG) improvement process based on grey level redistribution. Numerical experiments at multiple scales are performed for two different imposed displacement fields and results for one of the patterns generated are compared with those obtained for a random pattern and a Perlin noise one. Results: The proposed pattern is shown to lead to DIC errors comparable to those found with the two others for the first scales, but has much greater robustness. More importantly, the pattern gen-
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