Related papers: On the number of common sites visited by N random …
We compute analytically the mean number of common sites, W_N(t), visited by N independent random walkers each of length t and all starting at the origin at t=0 in d dimensions. We show that in the (N-d) plane, there are three distinct…
We study the mean number of encounters up to time t, E_N(t), taking place in a subspace with dimension d* of a d-dimensional lattice, for N independent random walkers starting simultaneously from the same origin. E_N is first evaluated…
In this Chapter, we consider a model of $N$ independent random walkers, each of duration $t$, and each starting from the origin, on a lattice in $d$ dimensions. We focus on two observables, namely $D_N(t)$ and $C_N(t)$ denoting respectively…
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…
We study the number of distinct sites S_N(t) and common sites W_N(t) visited by N independent one dimensional random walkers, all starting at the origin, after t time steps. We show that these two random variables can be mapped onto extreme…
Random walks on discrete lattices are fundamental models that form the basis for our understanding of transport and diffusion processes. For a single random walker on complex networks, many properties such as the mean first passage time and…
We present an analytical approach to study simple symmetric random walks (RWs) on a crossing geometry consisting of a plane square lattice crossed by $n_l$ number of lines that all meet each other at a single point (the origin) on the…
The evaluation of the average number S_N(t) of distinct sites visited up to time t by N independent random walkers all starting from the same origin on an Euclidean lattice is addressed. We find that, for the nontrivial time regime and for…
An overview is presented of recent work on some statistical problems on multiparticle random walks. We consider a Euclidean, deterministic fractal or disordered lattice and N >> 1 independent random walkers initially (t=0) placed onto the…
For a symmetric random walk in $Z^2$ which does not necessarily have bounded jumps we study those points which are visited an unusually large number of times. We prove the analogue of the Erd\H{o}s-Taylor conjecture and obtain the…
The set of visited sites and the number of visited sites are two basic properties of the random walk trajectory. We consider two independent random walks on a hyper-cubic lattice and study ordering probabilities associated with these…
We address the problem of evaluating the number $S_N(t)$ of distinct sites visited up to time t by N noninteracting random walkers all initially placed on one site of a deterministic fractal lattice. For a wide class of fractals, of which…
The average number $S_N(t)$ of distinct sites visited up to time t by N noninteracting random walkers all starting from the same origin in a disordered fractal is considered. This quantity $S_N(t)$ is the result of a double average: an…
Consider an arbitrary transient random walk on $\Z^d$ with $d\in\N$. Pick $\alpha\in[0,\infty)$ and let $L_n(\alpha)$ be the spatial sum of the $\alpha$-th power of the $n$-step local times of the walk. Hence, $L_n(0)$ is the range,…
The asymptotic mean number of distinct sites visited by a subdiffusive continuous time random walker in two dimensions seems not to have been explicitly calculated anywhere in the literature. This number has been calculated for other…
The exact mean time between encounters of a given particle in a system consisting of many particles undergoing random walks in discrete time is calculated, on both regular and complex networks. Analytical results are obtained both for…
A random walk starts from the origin of a d-dimensional lattice. The occupation number n(x,t) equals unity if after t steps site x has been visited by the walk, and zero otherwise. We study translationally invariant sums M(t) of observables…
For random walks on networks (graphs), it is a theoretical challenge to explicitly determine the mean first-passage time (MFPT) between two nodes averaged over all pairs. In this paper, we study the MFPT of random walks in the famous…
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…
We investigate random walks on complex networks and derive an exact expression for the mean first passage time (MFPT) between two nodes. We introduce for each node the random walk centrality $C$, which is the ratio between its coordination…