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

TL;DR

This paper explores the effects of various step size schemes on random walk behavior, revealing that certain shrinking step sizes lead to unique diffusion properties with an RMS displacement scaling as t^{1/4}.

Contribution

It introduces a novel variable step size scheme where the shrinking factor depends on the step number, extending previous fixed and geometric shrinking models.

Findings

Random step size and uniform shrinking are equivalent to fixed step size.

Geometric shrinking with fixed lambda exhibits distinct diffusion features.

Variable lambda_n leads to RMS displacement scaling as t^{1/4}.

Abstract

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.