On Properties of Non-Markovian Random Walk in One Dimension

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 $f(i)=e^{-v(i)}$.