Consider the random walk among N places with N(N - 1)/2 transports. We attach an exponential random variable Xij to each transport between places Pi and Pj and take these random variables mutually independent. If transports are possible or impossible independently with probability p and 1-p, respectively, then we give a lower bound for the distribution function of the smallest path at point log...

Recently, in [11], the “coin-turning walk” was introduced on Z. It is a non-Markovian process where steps form (possibly) time-inhomogeneous Markov chain. In this article, we follow up investigation by introducing analogous processes Zd,d≥2: at time n direction of “updated” with probability pn; otherwise next step repeats previous one. We study some fundamental properties these walks, such as t...