Similarity and self-similarity in random walk with fixed, random and shrinking steps

In this article, we first give a comprehensive description of random walk (RW) problem focusing on self-similarity, dynamic scaling and its connection to diffusion phenomena. One of the main goals of our work is to check how robust the RW problem is under various different choices of the step size. We show that RW with random step size or uniformly shrinking step size is exactly the same as for RW with fixed step size. Krapivsky and Redner in 2004 showed that RW with geometric shrinking step size, such that the size of the $n$th step is given by $S_n=\lambda^n$ with a fixed $\lambda<1$ value, exhibits some interesting features which are different from the RW with fixed step size. Motivated by this, we investigate what if $\lambda$ is not a fixed number rather it depends on the step number $n$? To this end, we first generate $N$ random numbers for RW of $t=N$ which are then arranged in a descending order so that the size of the $n$th step is $\lambda_n^n$. We have shown, both numerically and analytically, that $\lambda_n=(1-n/N)$, the root mean square displacement increases as $t^{1/4}$ which are different from all the known results on RW problems.