Contour lines of the discrete scale invariant rough surfaces

Abstract: We study the fractal properties of the 2d discrete scale invariant (DSI) rough surfaces. The contour lines of these rough surfaces show clear DSI. In the appropriate limit the DSI surfaces converge to the scale invariant rough surfaces. The fractal properties of the 2d DSI rough surfaces apart from possessing the discrete scale invariance property follow the properties of the contour lines of the corresponding scale invariant rough surfaces. We check this hypothesis by calculating numerous fractal exponents of the contour lines by using numerical calculations. Apart from calculating the known scaling exponents some other new fractal exponents are also calculated.

• The paper presents a numerical verification that critical scaling exponents for contour lines remain universal and depend solely on the Hurst exponent H.
• It employs large-scale simulations of 2D Weierstrass-Mandelbrot functions to measure fractal dimensions, loop distribution, and correlation exponents.
• Discrete scale invariance is shown to induce log-periodic modulations in contour geometry without altering the universal algebraic scaling laws.

Contour Lines of the Discrete Scale Invariant Rough Surfaces

Introduction and Context

This study investigates the fractal geometry of the contour lines arising from two-dimensional rough surfaces exhibiting discrete scale invariance (DSI). While scale invariance and related phenomena have been extensively explored in the context of critical phenomena and surface physics, DSI, which introduces log-periodic modulations in conventional power laws, remains less systematically characterized in the geometry of random surfaces. The Weierstrass-Mandelbrot (WM) function, a canonical example of a random continuous but non-differentiable function possessing DSI, serves as the mathematical foundation. The work primarily focuses on numerically measuring the scaling exponents of contour lines generated from 2D generalizations of the WM function and systematically testing the validity and universality of established scaling relations—especially their dependence on the Hurst roughness exponent $H$—for DSI, as opposed to continuously scale-invariant, surfaces.

Weierstrass-Mandelbrot Surfaces and Discrete Scale Invariance

The WM function is a mono-fractal process exhibiting DSI, where scaling invariance holds only for a discrete set of dilation factors. The 1D WM function is constructed as a sum over frequencies $\gamma^n$ with power-law weights. The 2D generalization, as formalized in Ausloos and Berman (1985), involves summing over angular and radial harmonics with random phases, leading to height fields %%%%2%%%% on a plane that are continuous, nowhere-differentiable, isotropic (for specific parameter choices), and DSI in their statistics. The Hurst parameter $H$ controls the global roughness; as $H$ increases, surfaces become smoother.

Critical to the study is the limit $\gamma\to 1$, in which WM surfaces converge to continuous scale-invariant rough surfaces such as the Brownian sheet, providing the opportunity to contrast DSI and continuous scale invariance.

Scaling Laws for Contour Lines

Contour lines, defined as height isolevels $h(x,y)=h_0$, form the principal objects of analysis. Their statistical and geometrical properties are encapsulated by several scaling exponents:

• Fractal dimension of individual contours $D_f$
• Fractal dimension of the full level set ensemble $d$
• Loop correlation exponent $x_l$ (governing the probability that two points at distance $r$ lie on the same contour)
• Length distribution exponent $\tau$ (which controls the power-law decay of loop perimeters)
• Two-point function exponent $n$ (in $G_s(r)$, the probability that two points at distance $r$ belong to a contour of fixed perimeter $s$)

The study builds upon theoretical scaling relations previously established for scale-invariant surfaces, such as:

• Hyperscaling: $D_f(\tau-1) = 2 - H$
• Sum rule: $D_f(3-\tau) = 2 - 2x_l$

For generic mono-fractal fields, the expectation is that all scaling exponents except $n$ depend solely on $H$.

Numerical Analysis and Verification of Scaling Relations

Large-scale simulations of 2D WM surfaces are performed for a range of $H$ values and for different choices of the generating periodic function $G(x)$, ensuring that statistical properties are not artifacts of particular functional forms. Surface sizes up to $2048 \times 2048$ account for finite size effects. Standard contouring algorithms are applied to extract ensembles of non-intersecting loops at various height levels.

Key findings include:

• DSI in Contour Geometry: The Lomb normalized periodogram applied to the radius-of-gyration sequence around loop centroids robustly detects discrete frequencies, confirming the DSI character of the contour geometry. The measured frequency ratios precisely match the construction parameter $\gamma$.
• Loop Correlation Exponent: Numerically, $x_l \approx 0.5$ for all $H$, confirming superuniversality—a result in concordance with continuously scale-invariant mono-fractal fields.
• Two-Point Function Exponent $n$: Unlike other exponents, $n$ varies linearly with $H$ as $n \approx (1+H)/2$. This scaling had previously not been theoretically predicted or measured; the empirical law holds across the studied range.
• Fractal Dimensions $D_f$ and $d$: Multiple independent estimators (length-radius relation, area cumulative statistics, and Zipf's law for ranked loops) all show excellent agreement with the theoretical predictions. Explicitly, $d = 2-H$ and $D_f$ matches analytic expressions derived from scaling relations.
• Length Distribution Exponent $\tau$: Measured via cumulative statistics to reduce noise, $\tau$ closely tracks the predicted scaling relations, with no significant dependence on $G(x)$.

All results are statistically indistinguishable between the DSI WM surfaces and their continuous scale-invariant limits, confirming that discrete scale invariance does not modify the universality classes of contour line exponents.

Implications and Further Directions

The findings substantiate the conjecture that for mono-fractal rough surfaces, the imposition of DSI leaves the critical exponents of geometric observables unchanged. This reflects a form of universality extending beyond the continuous scaling regime. The only detectable signature of DSI resides in log-periodic corrections and the appearance of discrete frequency components, not in the algebraic exponents.

From a practical perspective, the robustness of contour scaling exponents under DSI is significant for systems where natural or engineered roughness emerges from discrete hierarchical processes (e.g., in earth sciences, economics, or complex materials synthesis), as measurements on contours or isolevel curves can be interpreted within the broader paradigm established for mono-fractals.

Theoretically, the results reinforce the notion that scaling relations for contour lines on random fields—derived via arguments of self-similarity and hyperscaling—are fundamentally anchored in the roughness (Hurst) exponent, regardless of whether scale invariance is continuous or discrete. However, significant divergence is anticipated in the context of multi-fractal surfaces, where previous studies report large discrepancies, motivating further mathematical and numerical efforts.

A deeper understanding of DSI field theories, and the development of direct analytical tools (potentially extending to Schramm-Loewner Evolution in the DSI context), remains an open challenge. Additionally, devising observables that are sensitive to the distinction between DSI and full scale invariance on random fields could have important ramifications for the identification of universality classes in statistical mechanics and critical phenomena.

Conclusion

This work provides a comprehensive numerical investigation of the contour line geometry of discrete scale invariant rough surfaces, specifically using 2D Weierstrass-Mandelbrot functions. All canonical scaling exponents governing the ensemble of contours—fractal dimensions, distribution exponents, and correlation exponents—match those of continuously scale-invariant mono-fractal fields and are universal functions of the Hurst exponent. Discrete scale invariance manifests itself only through log-periodic modulations and the appearance of discrete harmonics in frequency space, not in the underlying critical exponents. These results solidify the theoretical framework for DSI in the geometry of random fields and delineate the boundaries of universality for contour statistics. Future work is required to systematically probe the distinctions arising in multifractal fields and to explore the implications for discrete scale invariant field theories.