Related papers: Random walk through a fertile site

Consider a one dimensional simple random walk $X=(X_n)_{n\geq0}$. We form a new simple symmetric random walk $Y=(Y_n)_{n\geq0}$ by taking sums of products of the increments of $X$ and study the two-dimensional walk…

We analyze the Brownian Motion limit of a prototypical unit step reinforced random-walk on the half line. A reinforced random walk is one which changes the weight of any edge (or vertex) visited to increase the frequency of return visits.…

We study the critical centered branching random walk with offspring and displacement distributions having finite variance, under minimal assumptions on its structure. We show that the probability that the position of the right-most particle…

We study the random walk problem on a class of deterministic Scale-Free networks displaying a degree sequence for hubs scaling as a power law with an exponent $\gamma=\log 3/\log2$. We find exact results concerning different first-passage…

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 $t_1$, $t_2$, $t_3$,…

Let $h:[0,1]\to\mathbb{R}$ be $C^2$ and such that $\sup_{[0,1]} h''<0$. For a (large) positive integer $n$, set $h_n(k) = n h(k/n)$ for any $k\in\{0,\dots,n\}$. We consider a random walk $(S_k)_{k\geq 0}$ with i.i.d.\ centred increments…

The integer points (sites) of the real line are marked by the positions of a standard random walk. We say that the set of marked sites is weakly, moderately or strongly sparse depending on whether the jumps of the standard random walk are…

We consider a discrete-time random walk on the nodes of an unbounded hexagonal lattice. We determine the probability generating functions, the transition probabilities and the relevant moments. The convergence of the stochastic process to a…

We investigate a self-interacting random walk, whose dynamically evolving environment is a random tree built by the walker itself, as it walks around. At time $n=1,2,\dots$, right before stepping, the walker adds a random number (possibly…

We study the mixing time of a random walker who moves inside a dynamical random cluster model on the d-dimensional torus of side-length n. In this model, edges switch at rate \mu between open and closed, following a Glauber dynamics for the…

We prove the limit theorem for paths of random walks with $n$ steps in $\mathbb{R}^d$ as $n$ and $d$ both go to infinity. For this, the paths are viewed as finite metric spaces equipped with the $\ell_p$-metric for $p\in[1,\infty)$. Under…

We consider a one-dimensional Brownian motion of fixed duration $T$. Using a path-integral technique, we compute exactly the probability distribution of the difference $\tau=t_{\min}-t_{\max}$ between the time $t_{\min}$ of the global…

In this paper we construct uniformly expanding random walks on smooth manifolds. In higher dimensions, our definition of uniform expansion measures the growth of subspaces rather than single vectors. Potrie showed that given any open set…

Let $G$ be a finitely generated group of polynomial volume growth equipped with a word-length $|\cdot|$. The goal of this paper is to develop techniques to study the behavior of random walks driven by symmetric measures $\mu$ such that, for…

The spatial coverage produced by a single discrete-time random walk, with asymmetric jump probability $p\neq 1/2$ and non-uniform steps, moving on an infinite one-dimensional lattice is investigated. Analytical calculations are complemented…

One goal in the quantum-walk research is the exploitation of the intrinsic quantum nature of multiple walkers, in order to achieve the full computational power of the model. Here we study the behaviour of two non-interacting particles…

We consider a random walk on a homogeneous space $G/\Lambda$ where $G$ is a non-compact simple Lie group and $\Lambda$ is a lattice. The walk is driven by a probability measure $\mu$ on $G$ whose support generates a Zariski-dense subgroup.…

Interactive programming environments are powerful tools for promoting innovative network thinking, teaching science of complexity, and exploring emergent phenomena. This paper reports on our recent development of the deterministic random…

Random walk on changing graphs is considered. For sequences of finite graphs increasing monotonically towards a limiting infinite graph, we establish transition probability upper bounds. It yields sufficient transience criteria for simple…

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…