Fractal sets arise naturally in analysis, it is one of the ways that mathematicians describe self-similarity. Self-similarity can be presented in a microscopic manner, for example, the Cantor set. However, we can work with self-similarity discretely in a macroscopic scale. Randomness can be added as well. For example, the images of lattice random walks.

The talk is divided into two parts. In the first part, we will go over notions of fractal dimensions. In the second part, I will introduce lattice random walks and their ranges. In particular, we will see that ranges of certain random walks are discrete fractals.

Measure theory is essential for this talk. Most of the talk will be based on the paper:
M.T. Barlow and S.J. Taylor. Defining fractal subsets of $\mathbb{Z}^d$. Proc. London. Math. Soc. (3), 64(1): 125-152, 1992.