Sharif Digital Repository

We study the fractal properties of the two-dimensional (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  

We investigate the statistics of isoheight lines of (2+1) -dimensional Kardar-Parisi-Zhang model at different level sets around the mean height in the saturation regime. We find that the exponent describing the distribution of the height-cluster size behaves differently for level cuts above and below the mean height, while the fractal dimensions of the height-clusters and their perimeters remain unchanged. The statistics of the winding angle confirms the previous observation that these contour lines are in the same universality class as self-avoiding random walks. © 2009 The American Physical Society  

In complex systems with fractal properties the scale invariance has an important rule to classify different statistical properties. In two dimensions the Loewner equation can classify all the fractal curves. Using the Weierstrass-Mandelbrot (WM) function as the drift of the Loewner equation we introduce a large class of fractal curves with discrete scale invariance (DSI). We show that the fractal dimension of the curves can be extracted from the diffusion coefficient of the trend of the variance of the WM function. We argue that, up to the fractal dimension calculations, all the WM functions follow the behavior of the corresponding Brownian motion. Our study opens a way to classify all the...