ABSTRACT
Fractal sets in nature are not only statistically self-similar, but are also always random. It may be said that, random fractal sets are certainly statistically self-similar where enlargements of small parts have the same statistical distribution as the whole set. A specific statistically self-similar construction is the randomized linear dusts presented by Mandelbrot and Chiles. Clearly, the one-dimensional random walk or Brownian random process is a self-affine fractal. This chapter discusses position function and fractal dimensions of the Brownian random process. An interesting application of an one-dimensional random walk to the fracture of materials has been considered by Takayasu. Percolation theory originates from the first paper of Broadbent & Hammersley. This theory satisfactorily describes many systems exhibiting a purely geometric phase transition. The concept of percolation can also be extended to microstructural elements of materials, such as dislocation lines, grain boundaries, and interfaces.
