Related papers: Vertex-Reinforced Random Walk
We consider random walks in which the walk originates in one set of nodes and then continues until it reaches one or more nodes in a target set. The time required for the walk to reach the target set is of interest in understanding the…
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…
In this work, we investigate the temporal evolution of the degree of a given vertex in a network by mapping the dynamics into a random walk problem in degree space. We analyze when the degree approximates a pre-established value through a…
Deterministic walks over a random set of points in one and two dimensions (d=1,2) are considered. Points (``cities'') are randomly scattered in R^d following a uniform distribution. A walker (a ``tourist''), at each time step, goes to the…
We consider a discrete-time Markovian random walk with resets on a connected undirected network. The resets, in which the walker is relocated to randomly chosen nodes, are governed by an independent discrete-time renewal process. Some nodes…
Consider a symmetric aperiodic random walk in $Z^d$, $d\geq 3$. There are points (called heavy points) where the number of visits by the random walk is close to its maximum. We investigate the local times around these heavy points and show…
We study asymptotic behaviours of a non-linear vertex-reinforced jump process defined on an arbitrary infinite graph with bounded degree. We prove that if the reinforcement function $w$ is reciprocally integrable and non-decreasing, then…
Given a non-negative Jacobi matrix describing higher order recurrence relations for multiple orthogonal polynomials of type~II and corresponding linear forms of type I, a general strategy for constructing a pair of stochastic matrices, dual…
This work deals with the stationary analysis of two-dimensional partially homogeneous nearest-neighbour random walks. Such type of random walks are characterized by the fact that the one-step transition probabilities are functions of the…
A random walk in $Z_+^2$ spatially homogeneous in the interior, absorbed at the axes, starting from an arbitrary point $(i_0,j_0)$ and with step probabilities drawn on Figure 1 is considered. The trivariate generating function of…
We consider continuous-time random walk models described by arbitrary sojourn time probability density functions. We find a general expression for the distribution of time-averaged observables for such systems, generalizing some recent…
We consider a random walk in a fixed Z environment composed of two point types: (q,1-q) and (p,1-p) for 1/2<q<p. We study the expected hitting time at N for a given number k of p-drifts in the interval [1,N-1], and find that this time is…
A step-reinforced random walk is a discrete-time stochastic process with long-range dependence. At each step, with a fixed probability $\alpha$, the so-called positively step-reinforced random walk repeats one of its previous steps, chosen…
We treat a quantum walk model with in- and out- flows at every time step from the outside. We show that this quantum walk can find the marked vertex of the complete graph with a high probability in the stationary state. In exchange of the…
We propose a model of random walks on weighted graphs where the weights are interval valued, and connect it to reversible imprecise Markov chains. While the theory of imprecise Markov chains is now well established, this is a first attempt…
Let $(\xi_1,\eta_1),(\xi_2,\eta_2),...$ be a sequence of i.i.d.\ copies of a random vector $(\xi,\eta)$ taking values in $\R^2$, and let $S_n := \xi_1+...+\xi_n$. The sequence $(S_{n-1} + \eta_n)_{n \geq 1}$ is then called perturbed random…
A Markov chain is considered whose states are orderings of an underlying fixed tree and whose transitions are local "random-to-front" reorderings, driven by a probability distribution on subsets of the leaves. The eigenvalues of the…
We give sharp, uniform estimates for the probability that a random walk of n steps on the reals avoids a half-line [y,infinity) given that it ends at the point x. The estimates hold for general continuous or lattice distributions provided…
Let $P$ be the transition matrix of a finite, irreducible and reversible Markov chain. We say the continuous time Markov chain $X$ has transition matrix $P$ and speed $\lambda$ if it jumps at rate $\lambda$ according to the matrix $P$. Fix…
We analyze time-discrete and continuous `fractional' random walks on undirected regular networks with special focus on cubic periodic lattices in $n=1,2,3,..$ dimensions. The fractional random walk dynamics is governed by a master equation…