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 N as Np is large.

In this study, we present a discrete-time non-Markov process, referred to as an Elephant Random Walk, implemented on an infinite one-dimensional lattice, with the inclusion of Random memory resetting. Upon each Random resetting event, the Walker completely loses its memory. Through analytical calculations, we determine the moments of displacement in the presence of Random resetting. Our findings demonstrate that the process does not attain a steady state. However, the long-time behavior of the moments reveals that, under specific conditions, the displacement distribution follows a Gaussian distribution. By manipulating the resetting mechanism, the transition from diffusive to superdiffusive behavior, or vice versa, can be induced in the process.

A discrete time multitype (p-type) branching Random Walk on the real line R is considered. The positions of the j-type individuals in the n-th generation form a point process. The asymptotic behavior of these point processes, when the generation size tends to infinity, is studied. The central limit theorem is proved

In this article, the famous Random Walk model is exploited as a model of stochastic processes to retrieve some specific words which are used in social media by users. By spreading activation on semantic networking, this model can predict the probability of the words’ activation, including all probabilities in different steps. In fact, the trend of probability in different steps is shown and the result of two different weights, when the steps tend to infinity is compared. In addition, it is shown that the results of the Random Walk model are aligned with the experimental psychological tests, showing that, as a model for semantic memory, it is a suitable model for retrieving in social media.

In this paper, we introduce a percolation model consisting of Random Walk movements on a lattice. Random Walk not only has local movements, but also has non-local movements on the lattice. We obtain the percolation transitions and critical exponents for this model. Our findings show that the percolation threshold decreases with increasing non-local movements. Also, we find the universal scaling functions for the size of the largest gap and biggest cluster by the extreme value theory.

Resetting in stochastic systems is defined in different ways. In this research, a 1D non-Markovian Random Walk is considered. In this process, the reset changes the dynamics in a way where the Random Walker, after losing its memory, goes back to a fixed point in space and starts again. In this study we investigate time evolution and also the long-time limit of displacement moments in the presence of resetting. Our calculations in the long-time limit show that the probability distribution function for displacement reaches a steady-state. On the other hand, this distribution never gets to a Gaussian form for any values of the resetting rate. We will show that, in contrast to the case where the process does not undergo resetting, the moments accumulate to finite values. This is confirmed by doing Monte Carlo simulations.