Related papers: No more than three favourite sites for simple rand…

It is shown that the path of a simple random walk on any graph, consisting of all vertices visited and edges crossed by the walk, is almost surely a recurrent subgraph.

We construct a bounded degree graph $G$, such that a simple random walk on it is transient but the random walk path (i.e., the subgraph of all the edges the random walk has crossed) has only finitely many cutpoints, almost surely. We also…

Random walks with memory typically involve rules where a preference for either revisiting or avoiding those sites visited in the past are introduced somehow. Such effects have a direct consequence on the statistics of first-passage and…

Benjamini,Haggstrom, Peres and Steif introduced the model of dynamical random walk on Z^d. This is a continuum of random walks indexed by a parameter t. They proved that for d=3,4 there almost surely exist t such that the random walk at…

We develop a framework to determine the complete statistical behavior of a fundamental quantity in the theory of random walks, namely, the probability that $n_1$, $n_2$, $n_3$, . . . distinct sites are visited at times $t_1$, $t_2$, $t_3$,…

We consider the problem, as suggested by Dembo ($2003$, $2006$), of late points of a simple random walk in two dimensions. It has been shown that the exponents for the numbers of pairs of late points coincide with those of nearly favorite…

Associated to a random walk on $\mathbb{Z}$ and a positive integer $n$, there is a return probability of the random walk returning to the origin after $n$ steps. An interesting question is when the set of return probabilities uniquely…

We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the…

In the classical paper of Dvoretzky-Erd\H{o}s, asymptotics for the expected value and the variance of the number of distinct sites visited by a Simple Symmetric Random Walk were calculated. Here, these results are generalized for Random…

In this paper we find an upper bound for the probability that a $3$ dimensional simple random walk covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_1$ ball of radius $N$. For $d\ge 4$, it has been shown in…

In this paper, we give a detailed construction of an example of excited random walk with speed zero in an ergodic random environment that have an infinite average number of cookies in each site. This example confirms that a result of…

We consider a transient excited random walk on $Z$ and study the asymptotic behavior of the occupation time of a currently most visited site. In particular, our results imply that, in contrast to the random walks in random environment, a…

The following random process on $\Z^4$ is studied. At first visit to a site, the two first coordinates perform a (2-dimensional) simple random walk step. At further visits, it is the last two coordinates which perform a simple random walk…

We introduce a class of nearest-neighbor integer random walks in random and non-random media, which includes excited random walks considered in the literature. At each site the random walker has a drift to the right, the strength of which…

We consider Sinai's random walk in random environment. We prove that infinitely often (i.o.) the size of the concentration neighborhood of this random walk is almost surely bounded. As an application we get that i.o. the maximal distance…

In a landscape composed of N randomly distributed sites in Euclidean space, a walker (``tourist'') goes to the nearest one that has not been visited in the last \tau steps. This procedure leads to trajectories composed of a transient part…

We survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time…

We consider a discrete time simple symmetric random walk on Z^d, d>=1, where the path of the walk is perturbed by inserting deterministic jumps. We show that for any time n and any deterministic jumps that we insert, the expected number of…

We show that the probability that a simple random walk covers a finite, bounded degree graph in linear time is exponentially small. More precisely, for every D and C, there exists a=a(D,C)>0 such that for any graph G, with n vertices and…

We consider the possible visits to visible points of a random walker moving up and right in the integer lattice (with probability $\alpha$ and $1-\alpha$, respectively) and starting from the origin. We show that, almost surely, the…