On Properties of Non-Markovian Random Walk in One Dimension

TL;DR

This paper investigates a non-Markovian one-dimensional random walk where the visit probability depends on previous visits, revealing diverse behaviors such as flat distributions, peaked distributions away from the origin, and different scaling of covering time based on the function of visits.

Contribution

It introduces and analyzes a class of non-Markovian random walks with visit-dependent probabilities, highlighting how different functions influence the walk's properties and scaling behavior.

Findings

For $f(i)=(v(i)+1)^{eta}$, the distribution tends to be flat for $eta>0$.

For $f(i)=e^{-v(i)}$, the walk develops two peaks away from the origin.

The lattice covering time scales as $N^{z}$ with $z=2$ for $eta o 0$, $z>2$ for $eta>0$, and $z<2$ for $f(i)=e^{-v(i)}$.

Abstract

We study a strongly Non-Markovian variant of random walk in which the probability of visiting a given site $i$ is a function $f$ of number of previous visits $v(i)$ to the site. If the probability is proportional to number of visits to the site, say $f(i)=(v(i)+1)^{\alpha}$ the probability distribution of visited sites tends to be flat for ${\alpha}>0$ compared to simple random walk. For $f(i)=e^{-v(i)}$, we observe a distribution with two peaks. The origin is no longer the most probable site. The probability is maximum at site k(t) which increases in time. For $f(i)=e^{-v(i)}$ and for ${\alpha}>0$ the properties do not change as the walk ages. However, for ${\alpha}<0$, the properties are similar to simple random walk asymptotically. We study lattice covering time for these functions. The lattice covering time scales as $N^{z}$, with $z=2$, for ${\alpha} \le 0$, $z>2$ for ${\alpha} >0$ and $z<2$ for $f(i)=e^{-v(i)}$.