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

For a simple symmetric random walk in dimension $d\geq 3$, a uniform strong law of large numbers is proved for the number of sites with given local time up to time $n$.

Motivated by the random Lorentz gas, we study deterministic walks in random environment and show that (in simple, yet relevant, cases) they can be reduced to a class of random walks in random environment where the jump probability depends…

We consider one dimensional random walks in random environment where every time the process stays at a location, it dies with a fixed probability. Under some mild assumptions it is easy to show that the survival probability goes to zero as…

In the present work, we study random walks on complex networks subject to stochastic resetting when the resetting probability is node-dependent. Using a renewal approach, we derive the exact expressions of the stationary occupation…

We study the excited random walk, in which a walk that is at a site that contains cookies eats one cookie and then hops to the right with probability p and to the left with probability q=1-p. If the walk hops onto an empty site, there is no…

This paper considers the question: how many times does a simple random walk revisit the most frequently visited site among the inner boundary points? It is known that in ${\mathbb{Z}}^2$, the number of visits to the most frequently visited…

The subject of this paper is the simple random walk on $\mathbb{Z}$. We give a very simple answer to the following problem: under the condition that a random walk has already spent $\alpha$-percent of the traveling time on the positive side…

The graph obtained from the integer grid Z x Z by the removal of all horizontal edges that do not belong to the x-axis is called a comb. In a random walk on a graph, whenever a walker is at a vertex v, in the next step it will visit one of…

We present an easy proof of Polya's theorem on random walks: with the probability one a random walk on the two-dimensional lattice returns to the starting point.

This paper considers 1-dimensional generalized random walks in random scenery. That is, the steps of the walk are generated by an arbitrary stationary process, and also the scenery is a priori arbitrary stationary. Under an ergodicity…

Random walks on graphs can be slow. To speed them up, imagine that at each step instead of choosing the neighbor at random, there is a small probability $\varepsilon>0$ that we can choose it. We show that in this case, at least for graphs…

We study a random walk in random environment on the non-negative integers. The random environment is not homogeneous in law, but is a mixture of two kinds of site, one in asymptotically vanishing proportion. The two kinds of site are (i)…

We prove that metric graph with the minimal growth of the number of possible endpoints of a random walk is the union of several linear paths coming out of the same vertex

We revisit an old minor topic in algorithms, the deterministic walk on a finite graph which always moves toward the nearest unvisited vertex until every vertex is visited. There is an elementary connection between this cover time and…

As the World Wide Web is growing rapidly, it is getting increasingly challenging to gather representative information about it. Instead of crawling the web exhaustively one has to resort to other techniques like sampling to determine the…

We investigate the minimal local time $f(n)$ of a one-dimensional simple random walk up to time $n$, defined as the smallest number of visits to any site in the range. A conjecture formulated repeatedly by Erd\H{o}s and R\'{e}v\'{e}sz…

For a one-dimensional simple random walk, let $g_1(n)$ denote the number of sites visited exactly once at time $n$. Major (1988) proved that \begin{equation*} \limsup_{n\to\infty}\frac{g_1(n)}{\log^2 n}=C\qquad a.s. \end{equation*} where…

Consider a random medium consisting of points randomly distributed so that there is no correlation among the distances. This is the random link model, which is the high dimensionality limit (mean field approximation) for the euclidean…

When providing directions to a place, web and mobile mapping services are all able to suggest the shortest route. The goal of this work is to automatically suggest routes that are not only short but also emotionally pleasant. To quantify…

We prove new results on lazy random walks on finite graphs. To start, we obtain new estimates on return probabilities $P^t(x,x)$ and the maximum expected hitting time $t_{\rm hit}$, both in terms of the relaxation time. We also prove a…