On properties of non-markovian random walk in one dimension

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 inversely proportional to number of visits to the site, say f(i)=1/(v(i)+1)α the probability distribution of visited sites tends to be flat for α>0 compared to simple random walk. For f(i)=exp(-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)=exp(-v(i)) and for α>0 the properties do not change as the walk ages. However, for α 2 for a >0 and z<2 for f(i)=exp(-v(i)).